HECKE-CLIFFORD SUPERALGEBRAS, CRYSTALS OF TYPE A_{2ℓ}^{(2)} AND MODULAR BRANCHING RULES FOR Ŝ_{n}

JONATHAN BRUNDAN AND ALEXANDER KLESHCHEV

1. Introduction

In [G1], Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (ℓ + 1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A_{ℓ}^{(1)}. His work yields amongst other things a new approach to the modular representations of the symmetric groups S_{n}, with the branching rules of [K1, K2, K3, B, BK1] being built into the theory from the outset. For a different approach, not followed here, we also cite the fundamental work of Ariki [A1, A2] and Ariki-Mathas [AM].

The present article is devoted to extending Grojnowski’s machinery to the twisted case: we replace the affine Hecke algebras with the affine Hecke-Clifford superalgebras of Jones and Nazarov [JN], and the Kac-Moody algebra A_{ℓ}^{(1)} with the twisted algebra 𝔤 = A_{2ℓ}^{(2)}. In particular, we obtain an algebraic construction purely in terms of the representation theory of Hecke-Clifford superalgebras of the plus part U_{ℤ}^{+} of the enveloping algebra of 𝔤, as well as of Kashiwara’s highest weight crystals B(∞) and B(λ) for each dominant weight λ.

The results of the article have applications to the modular representation theory of the double covers Ŝ_{n} of the symmeric groups, as was predicted originally by Leclerc and Thibon [LT], see also [BK2]. In particular, the parametrization of irreducibles, classification of blocks and analogues of the modular branching rules of the symmetric group for the double covers over fields of odd characteristic follow from the special case λ = Λ_{0} of our main results. These matters are discussed in the final section of the paper, §9.

Let us now describe the main results in more detail. Let H_{n} denote the affine Hecke-Clifford superalgebra of [JN], over an algebraically closed field F of characteristic different from 2 and at defining parameter a primitive (2ℓ + 1)-th root of unity q ∈ F^{×}. All results also have analogues in the degenerate case q = 1, working instead with the affine Sergeev superalgebra of [N], when the field F should be taken to be of characteristic (2ℓ + 1).

We consider

K(∞) = ⨁_{n≥0} K(Rep_{I} H_{n}),

the sum of the Grothendieck groups of integral ℤ_{2}-graded representations of H_{n} for all n (see §4-d for the precise definition). In a familiar way, K(∞) has a natural structure as a commutative graded Hopf algebra over ℤ, multiplication being induced by induction and comultiplication being induced by restriction. Hence, the graded dual K(∞)^{∗} is a cocommutative graded Hopf algebra over ℤ.

Next, let 𝔤 denote the twisted affine Kac-Moody algebra of type A_{2ℓ}^{(2)}. We will adopt standard Lie theoretic notation for the root system of 𝔤, summarized in more detail in the main body of the article. In particular, U_{ℚ} = U_{ℚ}^{−}U_{ℚ}^{0}U_{ℚ}^{+} denotes the ℚ-form of the universal enveloping algebra of 𝔤 generated by the Chevalley generators e_{i}, f_{i}, h_{i}(i ∈ I). Also U_{ℤ} = U_{ℤ}^{−}U_{ℤ}^{0}U_{ℤ}^{+} denotes the Kostant ℤ-form of U_{ℚ}. The first main theorem (Theorem 7.17) identifies K(∞)^{∗} with U_{ℤ}^{+}, viewing the latter as a graded Hopf algebra over ℤ via the principal grading:

2000 subject classification: 17B67, 20C08, 20C20, 17B10, 17B37.

Both authors partially supported by the NSF (grant nos DMS-9801442 and DMS-9900134).

Theorem A. K(∞)^{∗} and U_{ℤ}^{+} are isomorphic as graded Hopf algebras.

We also introduce for the first time for each dominant integral weight λ ∈ P_{+} a finite dimensional quotient superalgebra H_{n}^{λ} of H_{n}. These superalgebras play the role of the cyclotomic Hecke algebras in Grojnowski’s theory. In the special case λ = Λ_{0}, H_{n}^{λ} is the finite Hecke-Clifford superalgebra introduced by Olshanski [O]. Consider the sum of the Grothendieck groups

K(λ) = ⨁_{n≥0} K(Rep H_{n}^{λ})

of finite dimensional ℤ_{2}-graded H_{n}^{λ}-modules for all n. In this case, we can identify the graded dual K(λ)^{∗} with

K(λ)^{∗} = ⨁_{n≥0} K(Proj H_{n}^{λ})

where K(Proj H_{n}^{λ}) denotes the Grothendieck group of finite dimensional ℤ_{2}-graded projectives. It turns out that the natural Cartan map K(λ)^{∗} → K(λ) is injective (Theorem 7.10), so we can view both K(λ)^{∗} and K(λ) as lattices in K(λ)_{ℚ} = ℚ ⊗_{ℤ} K(λ) = ℚ ⊗_{ℤ} K(λ)^{∗}.

Each K(λ) has a natural structure of right K(∞)-comodule, with K(λ)^{∗} ⊂ K(λ) being a subcomodule. In other words, according to Theorem A, K(λ)^{∗} and K(λ) are left U_{ℤ}^{+}- modules. The action of the Chevalley generator e_{i} here is essentially a refinement of the restriction functors from H_{n}^{λ} to H_{n−1}^{λ}. We show moreover, by considering refinements of induction, that the action of U_{ℤ}^{+} extends to an action of all of U_{ℤ} on both K(λ) and K(λ)^{∗}. Hence, on extending scalars, we have an action of U_{ℚ} on K(λ)_{ℚ}, with K(λ)^{∗} ⊂ K(λ) being two different integral forms. The second main theorem (Theorem 7.16) is the following:

Theorem B. For each λ ∈ P_{+}, K(λ)_{ℚ} is the integrable highest weight U_{ℚ}-module of highest weight λ, with highest weight vector [1_{λ}] corresponding to the irreducible H_{0}^{λ}-module. Moreover K(λ)^{∗} ⊂ K(λ) are integral forms for K(λ)_{ℚ} with K(λ)^{∗} = U_{ℤ}^{−}[1_{λ}] and K(λ) being the dual lattice under the Shapovalov form.

Now let B(∞) denote the set of isomorphism classes of irreducible integral H_{n}-modules for all n ≥ 0. This has a natural crystal graph structure, the action of the crystal operators ẽ_{i} being defined by considering the socle of the restriction of an irreducible H_{n}-module to H_{n−1}. Similarly, writing B(λ) for the set of isomorphism classes of irreducible H_{n}^{λ}-modules for all n ≥ 0, each B(λ) has a natural crystal structure describing branching rules between the algebras H_{n}^{λ} and H_{n−1}^{λ}. We stress the crystal structures on B(∞) and each B(λ) are defined purely in terms of the representation theory of the Hecke-Clifford superalgebras. The next main result (Theorems 8.10 and 8.11) identifies the crystals:

Theorem C. The crystal B(∞) is isomorphic to Kashiwara’s crystal associated to the crystal base of U_{ℚ}^{−}. Moreover, for each λ ∈ P_{+}, the crystal B(λ) is isomorphic to Kashiwara’s crystal associated to the integrable highest weight U_{ℚ}-module of highest weight λ.

Acknowledgements. We would like to express our debt to the beautiful ideas of Ian Grojnowski in [G1]. Many of the proofs here, and certainly the overall strategy adopted in the article, are exactly as in Grojnowski’s work. We would also like to thank Monica Vazirani for explaining [GV] to us, in discussions which initiated the present work.

2. Affine Hecke-Clifford superalgebras

§2-a. Ground field and parameters. Let F be an algebraically closed field of characteristic different from 2, and choose q ∈ F^{×} such that

either q is a primitive odd root of unity (including the possibly q = 1), or q is not a root of unity at all.

Let h be the “quantum characteristic”, i.e. the smallest positive integer such that

q^{1−h} + q^{3−h} + ⋯ + q^{h−3} + q^{h−1} = 0

or ∞ in case no such integer exists. We refer to the case q ≠ 1 as the quantum case and q = 1 as the degenerate case. All the results will apply to either situation, but there are sufficiently many differences that for the purpose of exposition we will work in the quantum case in the main body of the text, with a summary of modifications in the degenerate case given at the end of each section whenever necessary. We set

ξ = q − q^{−1}

as a convenient shorthand.

§2-b. Modules over superalgebras. We will use freely the basic notions of superalgebra, referring the reader to [Le, ch.I], [Man, ch.3, §§1–2] and [BK2, §2]. We will denote the parity of a homogeneous vector v of a vector superspace by v̄ ∈ ℤ_{2}. By a superalgebra, we mean a ℤ_{2}-graded associative algebra over the fixed field F. If A and B are two superalgebras, then A ⊗ B = A ⊗_{F} B is again a superalgebra with multiplication satisfying

(a_{1} ⊗ b_{1})(a_{2} ⊗ b_{2}) = (−1)^{b̄_{1}ā_{2}}a_{1}a_{2} ⊗ b_{1}b_{2}

for a_{i} ∈ A, b_{i} ∈ B. Note this and other such expressions only make sense for homogeneous b_{1}, a_{2}: the intended meaning for arbitrary elements is to be obtained by extending linearly from the homogeneous case.

If A is a superalgebra, an A-module means a ℤ_{2}-graded left A-module. A morphism f : M → N of A-modules M and N means a (not necessarily homogeneous) linear map such that f(am) = (−1)f̄ ; āaf(m) for all a ∈ A, m ∈ M. The category of all such A-modules is denoted A-mod. By a submodule of an A-module, we always mean a graded submodule unless we explicitly say otherwise. We have the parity change functor

Π : A-mod → A-mod .  (2.1)

For an object M, ΠM is the same underlying vector space but with the opposite ℤ_{2}-grading. The new action of a ∈ A on m ∈ ΠM is defined in terms of the old action by a · m := (−1)āam.

It will occasionally be necessary to consider bimodules over two superalgebras A, B: an (A, B)-bimodule is an A-module M, as in the previous paragraph, which is also a ℤ_{2}- graded right B-module such that (am)b = a(mb) for all a ∈ A, b ∈ B, m ∈ M. Note a morphism f : M → N of (A, B)-bimodules means a morphism of A-modules as in the previous paragraph such that f(mb) = f(m)b for all m ∈ M, b ∈ B. This gives us the category A-mod-B of all (A, B)-bimodules. Also if M is an (A, B)-bimodule, ΠM denotes the A-module defined as in the previous paragraph, with the right B-action on ΠM being the same as the original action on M.

If M is a finite dimensional irreducible A-module, Schur’s lemma (e.g. [BK2, §2]) says that End_{A}(M) is either one dimensional, or two dimensional on basis id_{M}, θ_{M} where θ_{M} is an odd involution of M, unique up to a sign. In the former case, we call M an irreducible of type M, in the latter case M is an irreducible of type Q. We will occasionally write M ≃ N (as opposed to the usual M ≅ N) to indicate that there is actually an even isomorphism between A-modules M and N. For instance, if M is an irreducible of type Q, then M ≃ ΠM, while if M is irreducible of type M then M ≅ ΠM but M ≄ ΠM.

Given another superalgebra B and A-, B-modules M, N respectively, M ⊗ N = M ⊗_{F} N has a natural structure of A ⊗ B-module with

(a ⊗ b)(m ⊗ n) = (−1)^{b̄m̄}am ⊗ bn.

We will call this the outer tensor product of M and N and denote it M ⊠ N. If M and N are finite dimensional irreducibles, M ⊠ N need not be irreducible (unlike the purely even case). Indeed, if both M and N are of type M, then M ⊠N is also irreducible of type M, while if one is of type M and the other of type Q, then M ⊠ N is irreducible of type Q. In either of these two cases, it will be convenient to write M ⊛ N in place of M ⊠ N. But if both M and N are of type Q, then M ⊠ N is decomposable: let θ_{M} : M → M be an odd involution of M as an A-module and θ_{N} : N → N be an odd involution of N as a B-module. Then

θ_{M} ⊗ θ_{N} : M ⊠ N → M ⊠ N, m ⊗ n ↦ (−1)m̄θ_{M}(m) ⊗ θ_{N}(n)

is an even A⊗B-automorphism of M ⊠N whose square is −1. Therefore M ⊠N decomposes as a direct sum of two A ⊗ B-modules, namely, the ±\sqrt{−1}-eigenspaces of the linear map θ_{M} ⊗ θ_{N}. The map θ_{M} ⊗ id_{N} then gives an odd isomorphism between the two summands as A ⊗ B-modules. In this case, we pick either summand, and denote it M ⊛ N: it is an irreducible A ⊗ B-module of type M. Thus,

M ⊠ N ≃ \begin{cases}M ⊛ N ⊕ Π(M ⊛ N), & if M and N are both of type Q, \\ M ⊛ N, & otherwise. \end{cases}

We stress that M ⊛ N is in general only well-defined up to isomorphism.

Finally, we make some remarks about antiautomorphisms and duality. In this paper, all antiautomorphisms τ : A → A of a superalgebra A will be unsigned, so satisfy τ(ab) = τ(b)τ(a). If M is a finite dimensional A-module, then we can use τ to make the dual space M^{∗} = Hom_{F} (M, F) into an A-module by defining (af)(m) = f(τ(a)m) for all a ∈ A, f ∈ M^{∗}, m ∈ M. We will denote the resulting module by M^{τ}. We often use the fact that there is a natural even isomorphism

Hom_{A}(M, N) → Hom_{A}(N^{τ} , M^{τ})  (2.2)

for all finite dimensional A-modules M, N. The isomorphism sends f ∈ Hom_{A}(M, N) to the dual map f^{∗} ∈ Hom_{A}(N^{τ} , M^{τ}) defined by (f^{∗}θ)(m) = θ(fm) for all θ ∈ N^{τ}.

§2-c. Grothendieck groups. For a superalgebra A, the category A-mod is a superadditive category: each Hom_{A}(M, N) is a ℤ_{2}-graded abelian group in a way that is compatible with composition. Moreover, the underlying even category, i.e. the subcategory consisting of the same objects but only even morphisms, is an abelian category in the usual sense. This allows us to make use of all the basic notions of homological algebra on restricting our attention to even morphisms. For example by a short exact sequence in A-mod, we mean a sequence

0 ⟶ M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ 0  (2.3)

that is a short exact sequence in the underlying even category, so in particular all the maps are necessarily even. Note all functors between superadditive categories that we shall consider send even morphisms to even morphisms, so make sense on restriction to the underlying even subcategory.

Let us write Rep A for the full subcategory of A-mod consisting of all finite dimensional A-modules. We define the Grothendieck group K(Rep A) to be the quotient of the free ℤ-module with generators given by all finite dimensional A-modules by the submodule generated by

(1) M_{1} − M_{2} + M_{3} for every short exact sequence of the form (2.3);

(2) M − ΠM for every A-module M. We will write [M] for the image of the A-module M in K(Rep A). In an entirely similar way, we define the Grothendieck group K(Proj A), where Proj A denotes the full category of A-mod consisting of finite dimensional projectives.

Now suppose that A and B are two superalgebras. Then the Grothendieck groups K(Rep A) and K(Rep B) are free ℤ-modules with canonical bases corresponding to the isomorphism classes of irreducible modules. Moreover, there is an isomorphism

K(Rep A) ⊗_{ℤ} K(Rep B) → K(Rep A ⊗ B),  [L] ⊗ [L′] ↦ [L ⊛ L′]  (2.4)

for irreducible modules L ∈ Rep A, L′ ∈ Rep B. This simple observation explains the importance of the operation ⊛.

§2-d. The superalgebras. We now proceed to define the superalgebras we will be interested in. First, P_{n} denotes the algebra F [X_{1}^{±1}, …, X_{n}^{±n}] of Laurent polynomials, viewed as a superalgebra concentrated in degree ¯0. We record the relations:

X_{i}X_{i}^{−1} = 1, X_{i}^{−1}X_{i} = 1,  (2.5)
X_{k}X_{j} = X_{j}X_{k}, X_{k}^{−1}X_{j}^{−1} = X_{j}^{−1}X_{k}^{−1}, X_{k}X_{j}^{−1} = X_{j}^{−1}X_{k}, X_{k}^{−1}X_{j} = X_{j}X_{k}^{−1}  (2.6)

for all 1 ≤ i ≤ n, 1 ≤ j < k ≤ n. (Of course, some of these relations are redundant, but the precise form of the relations is used in the proof of Theorem 2.2.)

Let A_{n} denote the superalgebra with even generators X_{1}^{±1}, …, X_{n}^{±1} and odd generators C_{1}, …, C_{n}, where the X_{i}^{±1} are subject to the polynomial relations (2.5), (2.6), the C_{i} are subject to the Clifford superalgebra relations

C_{i}^{2} = 1,  (2.7)
C_{k}C_{j} = −C_{j}C_{k}  (2.8)

for all 1 ≤ i ≤ n, 1 ≤ j < k ≤ n, and there are the mixed relations

C_{i}X_{j} = X_{j}C_{i}, C_{i}X_{j}^{−1} = X_{j}^{−1}C_{i}, C_{i}X_{i} = X_{i}^{−1}C_{i}, C_{i}X_{i}^{−1} = X_{i}C_{i}  (2.9)

for all 1 ≤ i, j ≤ n with i ≠ j. For α = (α_{1}, …, α_{n}) ∈ ℤ^{n} and β = (β_{1}, …, β_{n}) ∈ ℤ_{2}^{n}, we write X^{α} and C^{β} for the monomials X_{1}^{α_{1}}X_{2}^{α_{2}} … X_{n}^{α_{n}} and C_{1}^{β_{1}}C_{2}^{β_{2}} … C_{n}^{β_{n}}, respectively. Then, it is straightforward to show that the elements

{X^{α}C^{β} | α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n}}

form a basis for A_{n}. In particular, A_{n} ≅ A_{1} ⊗ ⋯ ⊗ A_{1} (n times), and P_{n} can be identified with the subalgebra of A_{n} generated by the X_{i}^{±1}.

The symmetric group will be denoted S_{n}, with basic transpositions s_{1}, …, s_{n−1} and corresponding Bruhat ordering denoted ≤. We define a left action of S_{n} on A_{n} by algebra automorphisms so that

w · X_{i} = X_{wi},  w · C_{i} = C_{wi}  (2.10)

for each w ∈ S_{n}, i = 1, …, n.

Let H_{n}^{cl} denote the usual (classical) Hecke algebra of S_{n}, viewed as a superalgebra concentrated in degree ¯0. This can be defined on generators T_{1}, …, T_{n−1} subject to relations

T_{i}^{2} = ξT_{i} + 1,  (2.11)
T_{i}T_{j} = T_{j}T_{i}, T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}  (2.12)

for all admissible i, j with |i − j| > 2. Recall H_{n}^{cl} has a basis denoted {T_{w} | w ∈ S_{n}}, where T_{w} = T_{i_{1}} … T_{i_{m}} if w = s_{i_{1}} … s_{i_{m}} is any reduced expression for w. We also record

T_{i}^{−1} = T_{i} − ξ.  (2.13)

Now we are ready to define the main object of study: the affine Hecke-Clifford superalgebra H_{n}. This was first introduced by Jones and Nazarov [JN], being the q-analogue of the affine Sergeev superalgebra of [N]. By definition, H_{n} has even generators T_{1}, …, T_{n−1}, X_{1}^{±1}, …, X_{n}^{±1} and odd generators C_{1}, …, C_{n}, subject to the same relations as A_{n} (2.5), (2.6), (2.7), (2.8), (2.9) and as H_{n}^{cl} (2.11), (2.12), together with the new relations:

T_{i}C_{j} = C_{j}T_{i},  (2.14)
T_{i}C_{i} = C_{i+1}T_{i},  (2.15)
T_{i}X_{j} = X_{j}T_{i},  T_{i}X_{j}^{−1} = X_{j}^{−1}T_{i},  (2.16)
(T_{i} + ξC_{i}C_{i+1})X_{i}T_{i} = X_{i+1}  (2.17)

for all admissible i, j with j ≠ i, i + 1. The relation (2.15) is equivalent to

T_{i}C_{i+1} = C_{i}T_{i} − ξ(C_{i} − C_{i+1}),  (2.18)

while (2.17) is equivalent to any of the following four statements:

T_{i}X_{i} = X_{i+1}T_{i} − ξ(X_{i+1} + C_{i}C_{i+1}X_{i}),  (2.19)
T_{i}X_{i}^{−1} = X_{i+1}^{−1}T_{i} + ξ(X_{i}^{−1} + X_{i+1}^{−1}C_{i}C_{i+1}),  (2.20)
T_{i}X_{i+1} = X_{i}T_{i} + ξ(1 − C_{i}C_{i+1})X_{i+1},  (2.21)
T_{i}X_{i+1}^{−1} = X_{i}^{−1}T_{i} − ξX_{i}^{−1}(1 − C_{i}C_{i+1})  (2.22)

for all i = 1, …, n − 1. Using (2.19) and induction on j ≥ 1, one shows that

(T_{i} + ξC_{i}C_{i+1})X_{i}^{j} = X_{i+1}^{j}(T_{i} − ξ) − ξ ∑_{k=1}^{j−1} (X_{i}^{j−k}X_{i+1}^{k} + X_{i}^{−k}X_{i+1}^{j−k}C_{i}C_{i+1})  (2.23)

for all j ≥ 1, 1 ≤ i < n. Hence:

\begin{matrix}(T_{i} + ξC_{i}C_{i+1})X_{i}^{j}C_{i} = & X_{i+1}^{j}C_{i+1}(T_{i} + ξC_{i}C_{i+1}) \\ − ξ ∑_{k=1}^{j−1} (X_{i}^{j−k}X_{i+1}^{k}C_{i} − X_{i}^{−k}X_{i+1}^{j−k}C_{i+1}) , \end{matrix}  (2.24)

\begin{matrix}(T_{i} − ξ)X_{i}^{−j} = & X_{i+1}^{−j}(T_{i} + ξC_{i}C_{i+1}) \\ + ξ ∑_{k=1}^{j−1} (X_{i}^{−k}X_{i+1}^{k−j} + X_{i}^{j−k}X_{i+1}^{−k}C_{i}C_{i+1}) . \end{matrix}  (2.25)

Finally, let H_{n}^{fin} denote the subalgebra of H_{n} generated by all C_{i}, T_{j} for i = 1, …, n, j = 1, …, n − 1. Alternatively, as follows easily from Theorem 2.2 below, H_{n}^{fin} can be defined as the superalgebra generated by elements C_{i}, T_{j} subject only to the relations (2.7), (2.8), (2.11), (2.12) and (2.14), (2.15). Hence, H_{n}^{fin} is the (finite) Hecke-Clifford superalgebra first introduced by Olshanski [O] as the q-analogue of the Sergeev superalgebra of [S1].

§2-e. Basis theorem. Now we proceed to study the algebra H_{n} in more detail. The first goal is to construct a basis. There are obvious homomorphisms f : A_{n} → H_{n} and g : H_{n}^{cl} → H_{n} under which the X_{i}, C_{i} or T_{j} map to the same elements of H_{n}. We write X^{α}C^{β} also for the image under f of the basis element X^{α}C^{β} of A_{n}, and T_{w} for the image under g of T_{w} ∈ H_{n}^{cl}. This notation will be justified shortly, when we show that f and g are both algebra monomorphisms. The following lemma is obvious from the relations:

Lemma 2.1. Let f ∈ A_{n}, x ∈ S_{n}. Then, in H_{n},

T_{x}f = (x · f)T_{x} + ∑_{y<x} f_{y}T_{y}, fT_{x} = T_{x}(x^{−1} · f) + ∑_{y<x} T_{y}f′_{y}

for some f_{y}, f′_{y} ∈ A_{n}.

It follows easily that H_{n} is at least spanned by all X^{α}C^{β}T_{w} for all α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n} and w ∈ S_{n}. We wish to prove that these elements are linearly independent too:

Theorem 2.2. The {X^{α}C^{β}T_{w} | α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n}, w ∈ S_{n}} form a basis for H_{n}. Proof. Consider instead the algebra H̃_{n} on generators X̃_{i}^{±1}, C̃_{i}, T̃_{j} for 1 ≤ i ≤ n, 1 ≤ j < n subject to relations (2.5)–(2.9), (2.11), (2.14)–(2.16) and (2.18)–(2.22). Thus we have all the relations of H_{n} except for the braid relations (2.12). Using precisely these relations as the reduction system, it is a routine if tedious exercise using Bergman’s diamond lemma [Be, 1.2] to prove that H̃_{n} has a basis given by all X̃^{α}C̃^{β}T̃ for all α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n} and all words T̃ in the T̃_{j} which do not involve a subword of the form T̃_{j}^{2} for any j. Hence, the subalgebra Ã_{n} of H̃_{n} generated by the X̃_{i}^{±1}, C̃_{i} is isomorphic to A_{n}. Also let H̃_{n}^{cl} denote the subalgebra of H̃_{n} generated by the T̃_{j}, so that H̃_{n}^{cl} is isomorphic to the algebra on generators T̃_{1}, …, T̃_{n−1} subject to relations T̃_{j}^{2} = ξT̃_{j} + 1 for each j.

Now, by definition, H_{n} is the quotient of H̃_{n} by the two-sided ideal I generated by the elements

a_{i,j} = T̃_{i}T̃_{j} − T̃_{j}T̃_{i}, b_{i} = T̃_{i}T̃_{i+1}T̃_{i} − T̃_{i+1}T̃_{i}T̃_{i+1}

for i, j as in (2.12). Let J be the two-sided ideal of H̃_{n}^{cl} generated by the same elements a_{i,j}, b_{i} for all i, j. Then, by the basis theorem for H_{n}^{cl}, H̃_{n}^{cl}/J ≅ H_{n}^{cl}, with basis given by elements T_{w} for w ∈ S_{n} defined in the usual way. It follows immediately that to prove the theorem, it suffices to show that I = Ã_{n}J in H̃_{n}. In turn, this follows if we can show that rt = t′r for each r ∈ {a_{i,j}, b_{i}} and each generator t of Ã_{n}, where t′ is some other element of Ã_{n}. Now again this is a routine check: for example, the most complicated case involves verifying that (T̃_{i+1}T̃_{i}T̃_{i+1} − T̃_{i}T̃_{i+1}T̃_{i})X̃_{i+2} = X̃_{i}(T̃_{i+1}T̃_{i}T̃_{i+1} − T̃_{i}T̃_{i+1}T̃_{i}) using only the relations in H̃_{n}.

So we have a right from now on to identify A_{n} and H_{n}^{cl} with the corresponding subalgebras of H_{n}. The theorem also shows that the superalgebra H_{n}^{fin} has basis {C^{β}T_{w}|β ∈ ℤ_{2}^{n}, w ∈ S_{n}}. So Theorem 2.2 can be restated as saying that H_{n} is a free right H_{n}^{fin}-module on basis {X^{α} | α ∈ ℤ^{n}}.

As another consequence, it makes sense to consider the tower of superalgebras

H_{0} ⊂ H_{1} ⊂ H_{2} ⊂ ⋯ ⊂ H_{n} ⊂ …

where for i ≤ n, H_{i} is identified with the subalgebra of H_{n} generated by C_{1}, …, C_{i}, X_{1}^{±1}, …, X_{i}^{±1}, T_{1}, …, T_{i−1}. Similarly, we can consider A_{i} ⊆ A_{n}, P_{i} ⊆ P_{n}, etc....

Finally, we point out that there are obvious variants of the basis of Theorem 2.2, reordering the C’s, X’s and T ’s. For instance, H_{n} also has {T_{w}X^{α}C^{β} | w ∈ S_{n}, α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n}} as a basis. This follows using Lemma 2.1.

§2-f. The center of H_{n}. The next theorem was first established in [JN, Prop. 3.2] (for the case F = ℂ).

Theorem 2.3. The (super)center of H_{n} consists of all symmetric polynomials in X_{1} + X_{1}^{−1}, …, X_{n} + X_{n}^{−1}. Proof. Let Z denote the center of H_{n}, i.e. the z such that zy = yz for all y ∈ H_{n}. (The argument applies equally well to the supercenter defined by zy = (−1)^{z̄ȳ}yz.) One first checks that symmetric polynomials in X_{1} + X_{1}^{−1}, …, X_{n} + X_{n}^{−1} are central following the argument in the proof of [JN, Prop. 3.2(b)].

Conversely, take z = ∑_{w∈S_{n}} f_{w}T_{w} ∈ Z where each f_{w} ∈ A_{n}. Let w be maximal with respect to the Bruhat order such that f_{w} ≠ 0. Assume for a contradiction that w ≠ 1. Then, there exists some i ∈ {1, …, n} with wi ≠ i. Consider (X_{i} + X_{i}^{−1})z − z(X_{i} + X_{i}^{−1}). By Lemma 2.1, this looks like f_{w}(X_{i} + X_{i}^{−1} − X_{wi} − X_{wi}^{−1})T_{w} plus a linear combination of terms of the form f_{x}T_{x} for f_{x} ∈ A_{n} and x ∈ S_{n} with x ≱ w in the Bruhat order. So in view of Theorem 2.2, z is not central, a contradiction.

Hence, we must have that z ∈ A_{n}. Considering the form of the center of A_{n} one easily shows that z in fact lies in F [X_{1}+X_{1}^{−1}, …, X_{n}+X_{n}^{−1}]. To see that z is actually a symmetric polynomial, write z = ∑_{i,j≥0} a_{i,j}(X_{1} + X_{1}^{−1})^{i}(X_{2} + X_{2}^{−1})^{j} where the coefficients a_{i,j} lie in F [X_{3} + X_{3}^{−1}, …, X_{n} + X_{n}^{−1}]. Applying Lemma 2.1 to T_{1}z = zT_{1} now gives that a_{i,j} = a_{j,i} for each i, j, hence z is symmetric in X_{1} + X_{1}^{−1} and X_{2} + X_{2}^{−1}. Similar argument shows that z is symmetric in X_{i} + X_{i}^{−1} and X_{i+1} + X_{i+1}^{−1} for all i = 1, …, n − 1 to complete the proof.

§2-g. Parabolic subalgebras. Suppose that μ = (μ_{1}, …, μ_{u}) is a composition of n, i.e. a sequence of positive integers summing to n. Let S_{μ} ≅ S_{μ_{1}} × ⋯ × S_{μ_{u}} denote the corresponding Young subgroup of S_{n}, and H_{μ}^{cl} ⊆ H_{n}^{cl} denote its Hecke algebra. So H_{μ}^{cl} ≅ H_{μ_{1}}^{cl} ⊗ ⋯ ⊗ H_{μ_{u}}^{cl} is the subalgebra of H_{n}^{cl} generated by the T_{j} for which s_{j} ∈ S_{μ}.

We define the parabolic subalgebra H_{μ} of the affine Hecke-Clifford superalgebra H_{n} in a similar way: it is the subalgebra of H_{n} generated by A_{n} and all T_{j} for which s_{j} ∈ S_{μ}. It follows easily from Theorem 2.2 that the elements

{X^{α}C^{β}T_{w} | α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n}, w ∈ S_{μ}}

form a basis for H_{μ}. In particular, H_{μ} ≅ H_{μ_{1}} ⊗⋯⊗H_{μ_{u}}. Note that the parabolic subalgebra H_{(1,1,...,1)} is precisely the subalgebra A_{n}.

We will need the usual induction and restriction functors between H_{n} and H_{μ}. These will be denoted simply

ind_{μ}^{n} : H_{μ}-mod → H_{n}-mod,  res_{μ}^{n} : H_{n}-mod → H_{μ}-mod,  (2.26)

the former being the tensor functor H_{n}⊗_{H_{μ}}? which is left adjoint to res_{μ}^{n}. More generally, we will consider induction and restriction between nested parabolic subalgebras, with obvious notation. We will also occasionally consider the restriction functor

res_{n−1}^{n} : H_{n}-mod → H_{n−1}-mod  (2.27)

where H_{n−1} denotes the subalgebra of H_{n} generated by X_{i}^{±1}, C_{i} and T_{j} for i = 1, …, n − 1, j = 1, …, n − 2.

§2-h. Mackey theorem. Let μ, ν be compositions of n. We let D_{ν} denote the set of minimal length left S_{ν}-coset representatives in S_{n}, and D_{μ}^{−1} denote the set of minimal length right S_{μ}-coset representatives. Then D_{μ,ν} := D_{μ}^{−1} ∩ D_{ν} is the set of minimal length (S_{μ}, S_{ν})-double coset representatives in S_{n}. We recall some well-known properties, see e.g. [DJ, §1]. First, for x ∈ D_{μ,ν}, S_{μ} ∩ xS_{ν}x^{−1} and x^{−1}S_{μ}x ∩ S_{ν} are Young subgroups of S_{n}. So we can define compositions μ ∩ xν and x^{−1}μ ∩ ν of n from

S_{μ} ∩ xS_{ν}x^{−1} = S_{μ∩xν} and x^{−1}S_{μ}x ∩ S_{ν} = S_{x^{−1}μ∩ν}.

Then, for x ∈ D_{μ,ν}, every w ∈ S_{μ}xS_{ν} can be written as w = uxv for unique elements u ∈ S_{μ} and v ∈ S_{ν} ∩ D_{x^{−1}μ∩ν}^{−1}. Moreover, when this is done, ℓ(w) = ℓ(u) + ℓ(x) + ℓ(v). This fact implies the following well-known lemma which is essentially equivalent to the Mackey theorem for H_{n}^{cl} (see e.g. [DJ, Theorem 2.7]):

Lemma 2.4. For x ∈ D_{μ,ν}, the subspace H_{μ}^{cl}T_{x}H_{ν}^{cl} of H_{n}^{cl} has basis {T_{w} | w ∈ S_{μ}xS_{ν}}.

This is our starting point for proving a version of the Mackey theorem for H_{n}.

Lemma 2.5. For x ∈ D_{μ,ν}, the subspace H_{μ}T_{x}H_{ν}^{cl} of H_{n} has basis {X^{α}C^{β}T_{w} | α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n}, w ∈ S_{μ}xS_{ν}}. Moreover,

H_{n} = ⨁_{x∈D_{μ,ν}} H_{μ}T_{x}H_{ν}^{cl}.

Proof. Since H_{μ} = A_{n}H_{μ}^{cl}, Lemma 2.4 implies at once that the given {X^{α}C^{β}T_{w}} span H_{μ}T_{x}H_{ν}^{cl}. But they are linearly independent too by Theorem 2.2, proving the first statement. The second follows immediately using Theorem 2.2 once more.

Now fix some total order ≺ refining the Bruhat order < on D_{μ,ν}. For x ∈ D_{μ,ν}, set

B_{⪯x} = ⨁_{y∈D_{μ,ν}, y⪯x} H_{μ}T_{y}H_{ν}^{cl},  (2.28)
B_{≺x} = ⨁_{y∈D_{μ,ν}, y≺x} H_{μ}T_{y}H_{ν}^{cl},  (2.29)
B_{x} = B_{⪯x}/B_{≺x}.  (2.30)

It follows immediately from Lemma 2.1 that B_{⪯x} (resp. B_{≺x}) is invariant under right multiplication by A_{n}. Hence, since H_{ν} = H_{ν}^{cl}A_{n}, we have defined a filtration of H_{n} as an (H_{μ}, H_{ν})-bimodule. We want to describe the quotients B_{x} more explicitly as (H_{μ}, H_{ν})- bimodules.

Lemma 2.6. For each x ∈ D_{μ,ν}, there exists an algebra isomorphism

φ = φ_{x^{−1}} : H_{μ∩xν} → H_{x^{−1}μ∩ν}

with φ(T_{w}) = T_{x^{−1}wx}, φ(X_{i}) = X_{x^{−1}i}, and φ(C_{i}) = C_{x^{−1}i} for w ∈ S_{μ∩xν}, 1 ≤ i ≤ n. Proof. The isomorphism ψ : S_{μ∩xν} → S_{x^{−1}μ∩ν}, u ↦ x^{−1}ux is length preserving. Equivalently, x^{−1}(i + 1) = (x^{−1}i) + 1 for each i with s_{i} ∈ S_{μ∩xν}. Using this, it is straightforward to check that the map φ defined as above respects the defining relations on generators.

Let N be a left H_{x^{−1}μ∩ν}-module. By twisting the action with the isomorphism φ_{x^{−1}} : H_{μ∩xν} → H_{x^{−1}μ∩ν} from Lemma 2.6, we get a left H_{μ∩xν}-module, which will be denoted ^{x}N. Now we can identify the module B_{x} introduced above.

Lemma 2.7. View H_{μ} as an (H_{μ}, H_{μ∩xν})-bimodule and H_{ν} as an (H_{x^{−1}μ∩ν}, H_{ν})-bimodule in the natural ways. Then, ^{x}H_{ν} is an (H_{μ∩xν}, H_{ν})-bimodule and

B_{x} ≃ H_{μ} ⊗_{H_{μ∩xν}} ^{x}H_{ν}

as an (H_{μ}, H_{ν})-bimodule. Proof. We define a bilinear map H_{μ} × H_{ν} → B_{x} = B_{⪯x}/B_{≺x} by (u, v) ↦ uT_{x}v + B_{≺x}. For y ∈ S_{μ} ∩ xS_{ν}x^{−1},

T_{y}T_{x} = T_{yx} = T_{xx^{−1}yx} = T_{x}T_{x^{−1}yx}

which is all that is required to check that the map is H_{μ∩xν}-balanced. Hence there is an induced (H_{μ}, H_{ν})-bimodule map Φ : H_{μ} ⊗_{H_{μ∩xν}} ^{x}H_{ν} → B_{x}. Finally, to prove that Φ is bijective, note that

{X^{α}C^{β}T_{u} ⊗ T_{v} | α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n}, u ∈ S_{μ}, v ∈ S_{ν} ∩ D_{x^{−1}μ∩ν}^{−1}}

is a basis of the induced module H_{μ} ⊗_{H_{μ∩xν}} ^{x}H_{μ} as a vector space. In view of Lemma 2.5, the image of these elements under Φ is a basis of B_{x}.

Now we can prove the Mackey theorem, compare [G1, 5.2].

Theorem 2.8. (“Mackey Theorem”) Let M be an H_{ν}-module. Then res_{μ}^{n}ind_{ν}^{n}M admits a filtration with subquotients ≃ to ind_{μ∩xν}^{μ}^{x}(res_{x^{−1}μ∩ν}^{ν}M), one for each x ∈ D_{μ,ν}. Moreover, the subquotients can be taken in any order refining the Bruhat order on D_{μ,ν}, in particular ind_{μ∩ν}^{μ}res_{μ∩ν}^{ν}M appears as a submodule. Proof. This follows from Lemma 2.7 and the isomorphism

(H_{μ} ⊗_{H_{μ∩xν}} ^{x}H_{ν}) ⊗_{H_{ν}} M ≃ ind_{H_{μ∩xν}}^{H_{μ}}^{x}(res_{H_{x^{−1}μ∩ν}}^{H_{ν}}M),

which is easy to check.

§2-i. Some (anti)automorphisms. A check of relations shows that H_{n} posesses an automorphism σ and an antiautomorphism τ defined on the generators as follows:

σ : T_{i} ↦ −T_{n−i} + ξ,  C_{j} ↦ C_{n+1−j}, X_{j} ↦ X_{n+1−j};  (2.31)
τ : T_{i} ↦ T_{i} + ξC_{i}C_{i+1}, C_{j} ↦ C_{j},  X_{j} ↦ X_{j},  (2.32)

for all i = 1, …, n − 1, j = 1, …, n.

If M is a finite dimensional H_{n}-module, we can use τ to make the dual space M^{∗} into an H_{n}-module denoted M^{τ}, see §2-b. Note τ leaves invariant every parabolic subalgebra of H_{n}, so also induces a duality on finite dimensional H_{μ}-modules for each composition μ of n.

Instead, given any H_{n}-module M, we can twist the action with σ to get a new module denoted M^{σ}. More generally, for any composition ν = (ν_{1}, …, ν_{u}) of n we denote by ν^{∗} the composition with the same non-zero parts but taken in the opposite order. For example (3, 2, 1)^{∗} = (1, 2, 3). Then σ induces an isomorphism of parabolic subalgebras H_{ν^{∗}} → H_{ν}. So if M is an H_{ν}-module, we can inflate through σ to get an H_{ν^{∗}}-module denoted M^{σ}. If M = M_{1} ⊠ ⋯ ⊠ M_{u} is an outer tensor product module over H_{ν} then M^{σ} ≅ M_{u}^{σ} ⊠ ⋯ ⊠ M_{1}^{σ}. The same holds if each M_{i} is irreducible and ⊠ is replaced with ⊛. These observations imply:

Lemma 2.9. Let M ∈ H_{m}-mod and N ∈ H_{n}-mod. Then

(ind_{m,n}^{m+n}M ⊠ N)^{σ} ≅ ind_{n,m}^{m+n}N^{σ} ⊠ M^{σ}.

Moreover, if M and N are irreducible, the same holds for ⊛ in place of ⊠.

§2-j. Duality. Thoughout this subsection, let μ be a composition of n and set ν = μ^{∗}. Let d ∈ D_{μ,ν} be the longest double coset representative. Note that μ∩dν = μ and d^{−1}μ∩ν = ν, so S_{μ}dS_{ν} = S_{μ}d = dS_{ν}. There is an isomorphism

φ = φ_{d^{−1}} : H_{μ} → H_{ν},  (2.33)

see Lemma 2.6. As in §2-h, for an H_{ν}-module M, ^{d}M denotes the H_{μ}-module obtained by pulling back the action through φ. We begin by considering the classical situation, adopting the obvious analogous notation for modules over H_{μ}^{cl}, H_{ν}^{cl}.

Lemma 2.10. Define a linear map θ^{cl} : H_{n}^{cl} → ^{d}H_{ν}^{cl} by

θ^{cl}(T_{w}) = \{ \begin{matrix}T_{d^{−1}w} & if w ∈ dS_{ν}, \\ 0 & otherwise, \end{matrix}

for each w ∈ S_{n}. Then:

(i) θ^{cl} is an even homomorphism of (H_{μ}^{cl}, H_{ν}^{cl})-bimodules; (ii) ker θ^{cl} contains no non-zero left ideals of H_{n}^{cl}; (iii) the map

f^{cl} : H_{n}^{cl} → Hom_{H_{μ}^{cl}}(H_{n}^{cl},^{d}H_{ν}^{cl}), h ↦ hθ^{cl}

is an even isomorphism of (H_{n}^{cl}, H_{ν}^{cl})-bimodules. Proof. (i) Since d^{−1}μ∩ν = ν, H_{μ}^{cl}T_{d}H_{ν}^{cl} = T_{d}H_{ν}^{cl} is isomorphic as an (H_{μ}^{cl}, H_{ν}^{cl})-bimodule to dH_{ν}^{cl}, the isomorphism being simply the map T_{w} ↦ T_{d^{−1}w} for w ∈ dS_{ν}, compare Lemma 2.7. Now (i) follows because θ^{cl} is simply this isomorphism composed with the projection from H_{n}^{cl} to H_{μ}^{cl}T_{d}H_{ν}^{cl} along the bimodule decomposition H_{n}^{cl} = ⨁_{x∈D_{μ,ν}} H_{μ}^{cl}T_{x}H_{ν}^{cl}.

(ii) We show by downward induction on ℓ(x) that θ^{cl}(H_{n}^{cl}t) ≠ 0 whenever we are given x ∈ D_{ν} and

t = ∑_{y∈D_{ν} with ℓ(y)≤ℓ(x)} T_{y}h_{y}

with each h_{y} ∈ H_{ν}^{cl} and h_{x} ≠ 0. Since d is the longest element of D_{ν}, the induction starts with x = d: in this case, the conclusion is clear as θ^{cl}(t) = h_{x} ≠ 0. So now suppose x < d and that the claim has been proved for all higher x ∈ D_{ν}. Pick a basic transposition s such that sx > x and sx ∈ D_{ν}. Then,

T_{s}t = ∑_{y∈D_{ν} with ℓ(y)≤ℓ(sx),} T_{y}h′_{y}

for h′_{y} ∈ H_{ν}^{cl} with h′_{sx} = h_{x} ≠ 0. But now the induction hypothesis shows that θ^{cl}(H_{n}^{cl}t) = θ^{cl}(H_{n}^{cl}T_{s}t) ≠ 0.

(iii) We remind the reader that hθ^{cl} : H_{n}^{cl} → ^{d}H_{μ}^{cl} denotes the map with (hθ^{cl})(t) = (−1)^{h̄t̄}θ^{cl}(th) (the sign being + always in this case). Given this and (i), it is straightforward to check that f^{cl} is a homomorphism of (H_{n}^{cl}, H_{ν}^{cl})-bimodules. To see that it is an isomorphism, it suffices by dimension to show that it is injective. Suppose h lies in the kernel. Then, (f^{cl}(h))(t) = θ^{cl}(th) = 0 for all t ∈ H_{n}^{cl}. Hence h = 0 by (ii).

Now we extend this result to H_{n}, recalling the definition of φ from (2.33).

Lemma 2.11. Define a linear map θ : H_{n} → ^{d}H_{ν} by

θ(fT_{w}) = \{ \begin{matrix}φ(f)T_{d^{−1}w} & if w ∈ dS_{ν}, \\ 0 & otherwise, \end{matrix}

for each f ∈ A_{n}, w ∈ S_{n}. Then:

(i) θ is an even homomorphism of (H_{μ}, H_{ν})-bimodules; (ii) the map

f : H_{n} → Hom_{H_{μ}}(H_{n},^{d}H_{ν}), h ↦ hθ

is an even isomorphism of (H_{n}, H_{ν})-bimodules. Proof. (i) According to a special case of Lemma 2.7, the top factor B_{d} in the bimodule filtration of H_{n} defined in (2.30) is isomorphic to ^{d}H_{ν} as an (H_{μ}, H_{ν})-bimodule. The map θ is simply the composite of this isomorphism with the quotient map H_{n} → B_{d}.

(ii) Recall that {T_{w} | w ∈ D_{μ}^{−1}} forms a basis for H_{n} as a free left H_{μ}-module, and ^{d}H_{ν} is isomorphic to H_{μ} as a left H_{μ}-module. It follows that the maps

{ψ_{w} | w ∈ D_{μ}^{−1}}

form a basis for Hom_{H_{μ}}(H_{n},^{d}H_{ν}) as a free right H_{ν}-module, where ψ_{w} : H_{n} → ^{d}H_{ν} is the unique left H_{μ}-module homomorphism with ψ_{w}(T_{u}) = δ_{w,u}.1 for all u ∈ D_{μ}^{−1}.

The analogous maps ψ_{w}^{cl} ∈ Hom_{H_{μ}^{cl}}(H_{n}^{cl},^{d}H_{ν}^{cl}) defined by ψ_{w}^{cl}(T_{u}) = δ_{w,u}.1 for u ∈ D_{μ}^{−1} form a basis for Hom_{H_{μ}^{cl}}(H_{n}^{cl},^{d}H_{ν}^{cl}) as a free right H_{ν}^{cl}-module. So in view of Lemma 2.10(iii), we can find a basis {a_{w} |w ∈ D_{μ}^{−1}} for H_{n}^{cl} viewed as a right H_{ν}^{cl}-module such that f^{cl}(a_{w}) = ψ_{w}^{cl} for each w ∈ D_{μ}^{−1}, i.e.

θ^{cl}(T_{u}a_{w}) = \{ \begin{matrix}1 & if u = w, \\ 0 & otherwise \end{matrix}

for every u ∈ D_{μ}^{−1}. But H_{ν} = H_{ν}^{cl}A_{n}, so the elements {a_{w} | w ∈ D_{μ}^{−1}} also form a basis for H_{n} as a right H_{ν}-module, and f(a_{w}) = ψ_{w} since θ = θ^{cl} on H_{n}^{cl}. Thus f maps a basis of H_{n} to a basis of Hom_{H_{μ}}(H_{n},^{d}H_{ν}) (as free right H_{ν}-modules), hence f is an isomorphism of (H_{n}, H_{ν})-bimodules.

Corollary 2.12. There is a natural isomorphism Hom_{H_{μ}}(H_{n},^{d}M) ≃ H_{n} ⊗_{H_{ν}} M of H_{n}- modules, for every left H_{ν}-module M. Proof. Let f : H_{n} → Hom_{H_{μ}}(H_{n},^{d}H_{ν}) be the bimodule isomorphism constructed in Lemma 2.11. Then, there are natural isomorphisms

H_{n} ⊗_{H_{ν}} M ⟶^{f⊗id} Hom_{H_{μ}}(H_{n},^{d}H_{ν}) ⊗_{H_{ν}} M ≃ Hom_{H_{μ}}(H_{n},^{d}H_{ν} ⊗_{H_{ν}} M) ≃ Hom_{H_{n}}(H_{n},^{d}M),

the second isomorphism depending on the fact that H_{n} is a free left H_{μ}-module, see e.g. [AF, 20.10].

Recall the duality (2.32) on finite dimensional H_{n}- (resp. H_{ν}-) modules.

Corollary 2.13. There is a natural isomorphism ind_{ν}^{n}(M^{τ}) ≃ (ind_{μ}^{n}(^{d}M))^{τ} for every finite dimensional H_{ν}-module M. Proof. One routinely checks using (2.2) that the functor τ ∘ ind_{μ}^{n} ∘ τ (from the category of finite dimensional H_{μ}-modules to finite dimensional H_{n}-modules) is right adjoint to res_{μ}^{n}. Hence, it is isomorphic to Hom_{H_{μ}}(H_{n},?) by uniqueness of adjoint functors. Now combine this natural isomorphism with the previous corollary (with μ and ν swapped and d replaced by d^{−1}).

We finally record a special case, which is the analogue of [G1, Proposition 5.8]:

Theorem 2.14. Given a finite dimensional H_{m}-module M and a finite dimensional H_{n}- module N,

(ind_{m,n}^{m+n}M ⊠ N)^{τ} ≅ ind_{n,m}^{m+n}(N^{τ} ⊠ M^{τ}).

Moreover, if M and N are irreducible, the same is true with ⊠ replaced by ⊛.

§2-k. Modifications in the degenerate case. Now we summarize the necessary changes in the degenerate case q = 1. So now F is an algebraically closed field of odd characteristic h and q = 1. The superalgebras P_{n}, A_{n}, H_{n}, H_{μ}, H_{n}^{fin} need to be replaced with their degenerate analogues (but we keep the same symbols).

First of all, P_{n} becomes the ordinary polynomial algebra F [x_{1}, …, x_{n}] concentrated in degree ¯0. Then, A_{n} is replaced by the algebra on even generators x_{1}, …, x_{n} and odd generators c_{1}, …, c_{n}, where the x_{i} satisfy the relations of a polynomial ring and the c_{i} satisfy the same Clifford relations as before (2.7), (2.8). The relations (2.9) become instead:

c_{i}x_{i} = −x_{i}c_{i}, c_{i}x_{j} = x_{j}c_{i}  (2.34)

for 1 ≤ i, j ≤ n with i ≠ j. The algebra H_{n}^{cl} is replaced by the group algebra F S_{n} of the symmetric group, always writing simply w in place of T_{w} before.

Now the affine Hecke-Clifford superalgebra becomes the affine Sergeev superalgebra of [N]. This is the superalgebra H_{n} defined on even generators x_{1}, …, x_{n}, s_{1}, …, s_{n−1} and odd generators c_{1}, …, c_{n}. The relations between the x’s and c’s are as in the new A_{n}, the relations between the s_{i} are the usual relations of the symmetric group, i.e. (2.11), (2.12) with q there equal to 1, and there are new relations replacing (2.14)–(2.17):

s_{i}c_{i} = c_{i+1}s_{i},  s_{i}c_{i+1} = c_{i}s_{i},  s_{i}c_{j} = c_{j}s_{i},  (2.35)
s_{i}x_{i} = x_{i+1}s_{i} − 1 − c_{i}c_{i+1},  s_{i}x_{i+1} = x_{i}s_{i} + 1 − c_{i}c_{i+1},  s_{i}x_{j} = x_{j}s_{i}  (2.36)

for all admissible i, j with j ≠ i, i + 1. We record the useful formula, being the analogue of (2.23):

s_{i}x_{i}^{j} = x_{i+1}^{j}s_{i} − ∑_{k=0}^{j−1}(x_{i}^{j−k−1}x_{i+1}^{k} + x_{i}^{j−k−1}(−x_{i+1})^{k}c_{i}c_{i+1})  (2.37)

for all j ≥ 1, 1 ≤ i < n. Finally, H_{n}^{fin} is replaced by the subalgebra of H_{n} generated by the c_{i}, s_{j}: it is alternatively the twisted tensor product of a Clifford algebra with the group algebra of the symmetric group, i.e. the original Sergeev superalgebra considered in [S1], also see [BK2, §3].

The automorphism σ : H_{n} → H_{n} and antiautomorphism τ : H_{n} → H_{n} are now defined by:

σ : s_{i} ↦ −s_{n−i}, c_{j} ↦ c_{n+1−j}, x_{j} ↦ x_{n+1−j};  (2.38)
τ : s_{i} ↦ s_{i},  c_{j} ↦ c_{j},  x_{j} ↦ x_{j},  (2.39)

for all i = 1, …, n − 1, j = 1, …, n.

The basis theorem, proved in an analogous way to Theorem 2.2, now says that H_{n} has a basis given by {x^{α}c^{β}w | α ∈ ℤ_{≥0}^{n}, β ∈ ℤ_{2}^{n}, w ∈ S_{n}}. The center of H_{n}, proved as in Theorem 2.3, is now the set of all symmetric polynomials in x_{1}^{2}, x_{2}^{2}, …, x_{n}^{2} (see also [N, Prop. 3.1]). Parabolic subalgebras H_{μ} of H_{n} are defined in the same way as before. The Mackey Theorem and Theorem 2.14 are proved in an entirely similar way.

3. Cyclotomic Hecke-Clifford superalgebras

§3-a. Cyclotomic Hecke-Clifford superalgebras. Keep all the notation from the previous section. Suppose now that f ∈ F [X_{1}] ⊂ H_{n} is a polynomial of the form

a_{d}X_{1}^{d} + a_{d−1}X_{1}^{d−1} + ⋯ + a_{1}X_{1} + a_{0}  (3.1)

for coefficients a_{i} ∈ F with a_{d} = 1 and a_{i} = a_{0}a_{d−i} for each i = 0, 1, …, d. This assumption implies that

C_{1}f = a_{0}X_{1}^{−d}fC_{1}.  (3.2)

Define I_{f} to be the two-sided ideal of H_{n} generated by f and let

H_{n}^{f} := H_{n}/I_{f}.  (3.3)

We call H_{n}^{f} the cyclotomic Hecke-Clifford superalgebra corresponding to f.

§3-b. Basis theorem. The goal in this subsection is to describe an explicit basis for H_{n}^{f}. We introduce some further notation for the proof. Set f_{1} = f and for i = 2, …, n, define inductively f_{i} = (T_{i−1} +ξC_{i−1}C_{i})f_{i−1}T_{i−1}. The first lemma follows easily by induction from (2.23):

Lemma 3.1. For i = 1, …, n,

f_{i} = X_{i}^{d} + (terms lying in P_{i−1}X_{i}^{e}H_{i}^{fin} for 0 < e < d) + u_{i}

where u_{i} ∈ H_{i}^{fin} is a unit.

Given Z = {z_{1} < ⋯ < z_{u}} ⊆ {1, …, n}, let f_{Z} = f_{z_{1}}f_{z_{2}} … f_{z_{u}} ∈ H_{n}. Define

Π_{n} = {(α, Z) | Z ⊆ {1, …, n}, α ∈ ℤ^{n} with 0 ≤ α_{i} < d whenever i ∉ Z},  (3.4)
Π_{n}^{+} = {(α, Z) ∈ Π_{n} | Z ≠ ∅}.  (3.5)

Lemma 3.2. H_{n} is a free right H_{n}^{fin}-module on basis {X^{α}f_{Z} | (α, Z) ∈ Π_{n}}. Proof. Define a total order ≺ on ℤ so that

⌊\frac{d}{2}⌋ ≺ ⌊\frac{d}{2}⌋ − 1 ≺ ⌊\frac{d}{2}⌋ + 1 ≺ ⌊\frac{d}{2}⌋ − 2 ≺ ⌊\frac{d}{2}⌋ + 2 ≺ ….

We have a corresponding reverse lexicographic ordering on ℤ^{n}: α ≺ α′ if and only if α_{n} = α′_{n}, …, α_{k+1} = α′_{k+1}, α_{k} ≺ α′_{k} for some k = 1, …, n. It is important that ℤ^{n} has a smallest element with respect to this total order. Define a function γ : Π_{n} → ℤ^{n} by γ(α, Z) := (γ_{1}, …, γ_{n}) where

γ_{i} = \{ \begin{matrix}α_{i} & if i ∉ Z or α_{i} < 0, \\ α_{i} + d & if i ∈ Z and α_{i} ≥ 0. \end{matrix}

We claim that for (α, Z) ∈ Π_{n},

X^{α}f_{Z} = X^{γ(α,Z)}u + (terms lying in X^{β}H_{n}^{fin} for β ≺ γ(α, Z)),

where u is some unit in H_{n}^{fin}. In other words, X^{α}f_{Z} = X^{γ(α,Z)}u+(lower terms). Since γ : Π_{n} → ℤ^{n} is a bijection and we already know that the {X^{α} | α ∈ ℤ^{n}} form a basis for H_{n} viewed as a right H_{n}^{fin}-module by Theorem 2.2, the claim immediately implies the lemma.

To prove the claim, proceed by induction on n. If n = 1, the statement is quite obvious. Now assume n > 1 and the statement has been proved for (n − 1). We need to consider X^{α}f_{Z} for (α, Z) ∈ Π_{n}. If n ∉ Z, then the conclusion follows without difficulty from the induction hypothesis, so assume n ∈ Z. Let α′ = (α_{1}, …, α_{n−1}) and Z′ = Z − {n}. Then by induction

X^{α′}f_{Z′} = X^{γ(α′,Z′)}u′ + (terms lying in X^{β′}H_{n−1}^{fin} for β′ ≺ γ(α′, Z′)),

where u′ is some unit in H_{n−1}^{fin}. Multiplying on the left by X_{n}^{α_{n}} and on the right by f_{n}, using Lemma 3.1, one deduces that

X^{α}f_{Z} = X_{n}^{α_{n}+d}X^{γ(α′,Z′)}u′ + X_{n}^{α_{n}}X^{γ(α′,Z′)}u′u_{n} + (terms lying in X^{β}H_{n}^{fin} for β ≺ γ(α, Z))

where both u′ and u′u_{n} are units in H_{n}^{fin}. Noting that

X^{γ(α,Z)} = \{ \begin{matrix}X_{n}^{α_{n}+d}X^{γ(α′,Z′)} & if α_{n} ≥ 0, \\ X_{n}^{α_{n}}X^{γ(α′,Z′)} & if α_{n} < 0, \end{matrix}

this is of the desired form.

Lemma 3.3. (i) For 1 ≤ i ≤ n,

C_{i}f_{i}H_{n}^{fin} ⊆ X_{i}^{−d}f_{i}H_{n}^{fin} + ∑_{k=1}^{i−1} P_{i}f_{k}H_{n}^{fin}.

(ii) For 1 ≤ i ≤ j ≤ n,

P_{j}(T_{j−1} − ξ) … (T_{i+1} − ξ)(T_{i} − ξ)X_{i}^{−d}f_{i}H_{n}^{fin} ⊆ ∑_{k=1}^{j} P_{j}f_{k}H_{n}^{fin}.

(iii) For n > 1, H_{n−1}^{fin}f_{n}H_{n}^{fin} = f_{n}H_{n}^{fin}. Proof. (i) Proceed by induction on i, the case i = 1 being immediate from (3.2). For i > 1,

C_{i}f_{i}H_{n}^{fin} = C_{i}(T_{i−1} + ξC_{i−1}C_{i})f_{i−1}H_{n}^{fin} = (T_{i−1} − ξ)C_{i−1}f_{i−1}H_{n}^{fin}

⊆ (T_{i−1} − ξ)X_{i−1}^{−d}f_{i−1}H_{n}^{fin} + ∑_{k=1}^{i−2}(T_{i−1} − ξ)P_{i−1}f_{k}H_{n}^{fin}

⊆ X_{i}^{−d}(T_{i−1} + ξC_{i−1}C_{i})f_{i−1}H_{n}^{fin} + ∑_{k=1}^{i−1} P_{i}f_{k}H_{n}^{fin} = X_{i}^{−d}f_{i}H_{n}^{fin} + ∑_{k=1}^{i−1} P_{i}f_{k}H_{n}^{fin},

applying Lemma 2.1, the relations in H_{n} especially (2.25), and the induction hypothesis.

(ii) Proceed by induction on (j − i), the conclusion being immediate in case j = i. If j > i, an application of (2.25), combined with (i) to commute C_{i} past f_{i}, gives that

P_{j}(T_{j−1} − ξ) … (T_{i} − ξ)X_{i}^{−d}f_{i}H_{n}^{fin} ⊆ P_{j}(T_{j−1} − ξ) … (T_{i+1} − ξ)X_{i+1}^{−d}f_{i+1}H_{n}^{fin} + ∑_{k=1}^{j} P_{j}f_{k}H_{n}^{fin}.

Now apply the induction hypothesis.

(iii) By considering the antiautomorphism τ of H_{n−1}^{fin}, one sees that H_{n−1}^{fin} is generated by the elements C_{i} for 1 ≤ i ≤ n−1 and (T_{j} +ξC_{j}C_{j+1}) for 1 ≤ j < n−1, and the latter satisfy the braid relations. We show that each of these generators of H_{n−1}^{fin} leave f_{n}H_{n}^{fin} invariant.

First, consider C_{i}f_{n}H_{n}^{fin} for 1 ≤ i ≤ n−1. Expanding the definition of f_{n} and commuting C_{i} past the leading terms, it equals

(T_{n−1} + ξC_{n−1}C_{n}) … C_{i}(T_{i} + ξC_{i}C_{i+1})f_{i}H_{n}^{fin}.

By the relations, C_{i}(T_{i} + ξC_{i}C_{i+1}) = (T_{i} + ξC_{i}C_{i+1})C_{i+1}. Now the conclusion in this case follows immediately since C_{i+1}f_{i}H_{n}^{fin} = f_{i}H_{n}^{fin}.

Next, consider (T_{j} + ξC_{j}C_{j+1})f_{n}H_{n}^{fin} for 1 ≤ j < n − 1. Expanding and commuting again gives that it equals

(T_{n−1} + ξC_{n−1}C_{n}) … (T_{j} + ξC_{j}C_{j+1})(T_{j+1} + ξC_{j+1}C_{j+2})(T_{j} + ξC_{j}C_{j+1})f_{j}H_{n}^{fin}.

Applying the braid relation gives

(T_{n−1} + ξC_{n−1}C_{n}) … (T_{j+1} + ξC_{j+1}C_{j+2})(T_{j} + ξC_{j}C_{j+1})(T_{j+1} + ξC_{j+1}C_{j+2})f_{j}H_{n}^{fin}

which again equals f_{n}H_{n}^{fin} as required.

Lemma 3.4. I_{f} = ∑_{i=1}^{n} P_{n}f_{i}H_{n}^{fin}.

Proof. Let H_{2...n}^{fin} denote the subalgebra of H_{n}^{fin} generated by C_{2}, …, C_{n}, T_{2}, …, T_{n−1}, so H_{2...n}^{fin} ≅ H_{n−1}^{fin}. We first claim that

H_{n}^{fin} ⊆ ∑_{i=1}^{n}(T_{i−1} + ξC_{i−1}C_{i}) … (T_{1} + ξC_{1}C_{2})H_{2...n}^{fin} + ∑_{i=1}^{n}(T_{i−1} − ξ) … (T_{1} − ξ)C_{1}H_{2...n}^{fin}.

To prove this, it suffices to show that T_{w}C^{β} lies in the right hand side for each w ∈ S_{n}, β ∈ ℤ_{2}^{n}. Proceed by induction on the Bruhat order on w ∈ S_{n}, the case w = 1 being trivial. For the induction step, we can find 1 ≤ i ≤ n and u ∈ S_{2...n} such that

T_{w}C^{β} = T_{i−1} … T_{1}T_{u}C^{β} = T_{i−1} … T_{1}C_{1}^{β_{1}}T_{u}C^{β′}

where β′ = (¯0, β_{2}, …, β_{n}). Now an argument using Lemma 2.1 gives that

T_{w}C^{β} = \{ \begin{matrix}(T_{i−1} + ξC_{i−1}C_{i}) … (T_{1} + ξC_{1}C_{2})T_{u}C^{β′} & if β_{1} = ¯0, \\ (T_{i−1} − ξ) … (T_{1} − ξ)C_{1}T_{u}C^{β′} & if β_{1} = ¯1, \end{matrix}

modulo lower terms of the form T_{w′}C^{β′′} with w′ < w. The claim follows.

Now let J = ∑_{i=1}^{n} P_{n}f_{i}H_{n}^{fin}. Clearly J ⊆ I_{f}. So it suffices for the lemma to show that I_{f} ⊆ J. Noting that H_{n} = P_{n}H_{n}^{fin} and using the result in the previous paragraph, we get that

I_{f} = H_{n}f_{1}H_{n} = H_{n}f_{1}P_{n}H_{n}^{fin} = H_{n}f_{1}H_{n}^{fin} = P_{n}H_{n}^{fin}f_{1}H_{n}^{fin}

⊆∑_{i=1}^{n}P_{n}(T_{i−1} + ξC_{i−1}C_{i}) … (T_{1} + ξC_{1}C_{2})f_{1}H_{n}^{fin}+∑_{i=1}^{n}P_{n}(T_{i−1} − ξ) … (T_{1} − ξ)C_{1}f_{1}H_{n}^{fin}.

Now, each P_{n}(T_{i−1} +ξC_{i−1}C_{i}) … (T_{1} +ξC_{1}C_{2})f_{1}H_{n}^{fin} = P_{n}f_{i}H_{n}^{fin} ⊆ J. Also each P_{n}(T_{i−1} − ξ) … (T_{1} − ξ)C_{1}f_{1}H_{n}^{fin} is contained in J thanks to Lemma 3.3(i),(ii).

Lemma 3.5. I_{f} = ∑_{(α,Z)∈Π_{n}^{+}} X^{α}f_{Z}H_{n}^{fin}.

Proof. We proceed by induction on n, the case n = 1 being almost obvious. So now suppose that n > 1. Clearly we can assume that d > 0. It will be convenient to write I′_{f} for the two-sided ideal of H_{n−1} generated by f, so

I′_{f} = ∑_{(α′,Z′)∈Π_{n−1}^{+}} X^{α′}f_{Z′}H_{n−1}^{fin}  (3.6)

by the induction hypothesis. Let J = ∑_{(α,Z)∈Π_{n}^{+}} X^{α}f_{Z}H_{n}^{fin}. Obviously J ⊆ I_{f}. So in view of Lemma 3.4, it suffices to show that X^{α}f_{i}H_{n}^{fin} ⊆ J for each α ∈ ℤ^{n} and each i = 1, …, n.

Consider first X^{α}f_{n}H_{n}^{fin}. Write X^{α} = X_{n}^{α_{n}}X^{β} for β ∈ ℤ^{n−1}. Expanding X^{β} in terms of the basis of H_{n−1} from Lemma 3.2, we see that

X^{α}f_{n}H_{n}^{fin} ⊆ ∑_{(α′,Z′)∈Π_{n−1}} X_{n}^{α_{n}}X^{α′}f_{Z′}H_{n−1}^{fin}f_{n}H_{n}^{fin}.

This is contained in J thanks to Lemma 3.3(iii).

Finally, consider X^{α}f_{i}H_{n}^{fin} with i < n. Write X^{α} = X_{n}^{α_{n}}X^{β} for β ∈ ℤ^{n−1}. By the induction hypothesis,

X^{α}f_{i}H_{n}^{fin} = X_{n}^{α_{n}}X^{β}f_{i}H_{n}^{fin} ⊆ ∑_{(α′,Z′)∈Π_{n−1}^{+}} X_{n}^{α_{n}}X^{α′}f_{Z′}H_{n}^{fin}.

Now we need to consider the cases α_{n} ≥ 0 and α_{n} < d separately. The argument is entirely similar in each case, so suppose in fact that α_{n} ≥ 0. Then we show by induction on α_{n} that X_{n}^{α_{n}}X^{α′}f_{Z′}H_{n}^{fin} ⊆ J for each (α′, Z′) ∈ Π_{n−1}^{+}. This is immediate if α_{n} < d, so take α_{n} ≥ d and consider the induction step. Expanding f_{n} using Lemma 3.1, the set

X_{n}^{α_{n}−d}X^{α′}f_{Z′}f_{n}H_{n}^{fin} ⊆ J

looks like the desired X_{n}^{α_{n}}X^{α′}f_{Z′}H_{n}^{fin} plus a sum of terms lying in X_{n}^{α_{n}−d+e}I′_{f}H_{n}^{fin} with 0 ≤ e < d. It now suffices to show that each such X_{n}^{α_{n}−d+e}I′_{f}H_{n}^{fin} ⊆ J. But by (3.6),

X_{n}^{α_{n}−d+e}I′_{f}H_{n}^{fin} ⊆ ∑_{(α′,Z′)∈Π_{n−1}^{+}} X_{n}^{α_{n}−d+e}X^{α′}f_{Z′}H_{n}^{fin}

and each such term lies in J by induction, since 0 ≤ α_{n} − d + e < α_{n}.

Theorem 3.6. The canonical images of the elements

{X^{α}C^{β}T_{w} | α ∈ ℤ^{n} with 0 ≤ α_{1}, …, α_{n} < d, β ∈ ℤ_{2}^{n}, w ∈ S_{n}}

form a basis for H_{n}^{f}.

Proof. By Lemmas 3.2 and 3.5, the elements {X^{α}f_{Z} | (α, Z) ∈ Π_{n}^{+}} form a basis for I_{f} viewed as a right H_{n}^{fin}-module. Hence Lemma 3.2 implies that the elements

{X^{α} | α ∈ ℤ^{n} with 0 ≤ α_{1}, …, α_{n} < d}

form a basis for a complement to I_{f} in H_{n} viewed as a right H_{n}^{fin}-module. The theorem follows at once.

§3-c. Cyclotomic Mackey theorem. We will need a special case of a Mackey theorem for cyclotomic Hecke-Clifford superalgebras. Let f ∈ P_{1} be a polynomial of degree d > 0 of the special form (3.1). Let H_{n}^{f} denote the corresponding cyclotomic Hecke-Clifford superalgebra. Given any y ∈ H_{n}, we will write ỹ for its canonical image in H_{n}^{f}. Thus, Theorem 3.6 says that the elements

{X̃^{α}C̃^{β}T̃_{w} | α ∈ ℤ^{n} with 0 ≤ α_{1}, …, α_{n} < d, β ∈ ℤ_{2}^{n}, w ∈ S_{n}}

form a basis for H_{n}^{f}.

It is obvious from Theorem 3.6 that the subalgebra of H_{n+1}^{f} spanned by the X̃^{α}C̃^{β}T̃_{w} for α ∈ ℤ^{n} with 0 ≤ α_{1}, …, α_{n} < d, β ∈ ℤ_{2}^{n} and w ∈ S_{n} is isomorphic to H_{n}^{f}. We will write

ind_{H_{n}^{f}}^{H_{n+1}^{f}} and res_{H_{n}^{f}}^{H_{n+1}^{f}} for the induction and restriction functors between H_{n}^{f} and H_{n+1}^{f}, to avoid confusion with the affine analogue from (2.27). So,

ind_{H_{n}^{f}}^{H_{n+1}^{f}}M = H_{n+1}^{f} ⊗_{H_{n}^{f}} M.

Lemma 3.7. For all 1 ≤ i < n, j ≥ 0, k ∈ ℤ_{2},

X_{i+1}^{j}C_{i+1}^{k}T_{i} − T_{i}X_{i}^{j}C_{i}^{k} ∈ ∑_{h=1}^{j} (X_{i+1}^{h}A_{i} + X_{i+1}^{h−1}C_{i+1}A_{i}) ,

Proof. Rearrange (2.23) and (2.24).

Lemma 3.8. (i) H_{n+1}^{f} is a free right H_{n}^{f}-module on basis

{X̃_{j}^{a}C̃_{j}^{b}T̃_{j} … T̃_{n} | 0 ≤ a < d, b ∈ ℤ_{2}, 1 ≤ j ≤ n + 1}.

(ii) H_{n+1}^{f} = H_{n}^{f}T̃_{n}H_{n}^{f} ⊕ ⨁_{0≤a<d,b∈ℤ_{2}} X̃_{n+1}^{a}C̃_{n+1}^{b}H_{n}^{f} as an (H_{n}^{f}, H_{n}^{f})-bimodule.

(iii) For any 0 ≤ a < d, there are isomorphisms

X̃_{n+1}^{a}H_{n}^{f} ≃ H_{n}^{f},  X̃_{n+1}^{a}C̃_{n+1}H_{n}^{f} ≃ ΠH_{n}^{f},  H_{n}^{f}T̃_{n}H_{n}^{f} ≃ H_{n}^{f} ⊗_{H_{n−1}^{f}} H_{n}^{f},

as (H_{n}^{f}, H_{n}^{f})-bimodules. Proof. For (i), by Theorem 3.6 and dimension considerations, we just need to check that H_{n+1}^{f} is generated as a right H_{n}^{f}-module by the given elements. For this, it suffices to show that all elements of the form

X̃^{α}C̃^{β}T̃_{j} … T̃_{n} (α ∈ ℤ^{n}, β ∈ ℤ_{2}^{n}, 0 ≤ α_{1}, …, α_{n+1} < d)

lie in the right H_{n}^{f}-module generated by {X̃_{k}^{a}C̃_{k}^{b}T̃_{k} … T̃_{n} | 0 ≤ a < d, b ∈ ℤ_{2}, j ≤ k ≤ n + 1}. This involves considering terms of the form X̃_{k}^{a}C̃_{k}^{b}T̃_{k−1} … T̃_{n} for j < k ≤ n + 1 and 0 ≤ a < d, b ∈ ℤ_{2}, for which Lemma 3.7 is useful.

For (ii),(iii), define a map H_{n}^{f} × H_{n}^{f} → H_{n}^{f}T̃_{n}H_{n}^{f}, (u, v) ↦ uT̃_{n}v. This is H_{n−1}^{f}-balanced, so induces a well-defined epimorphism

Φ : H_{n}^{f} ⊗_{H_{n−1}^{f}} H_{n}^{f} → H_{n}^{f}T̃_{n}H_{n}^{f}

of (H_{n}^{f}, H_{n}^{f})-bimodules. We know from (i) that H_{n}^{f} ⊗_{H_{n−1}^{f}} H_{n}^{f} is a free right H_{n}^{f}-module on

basis X̃_{j}^{a}C̃_{j}^{b}T̃_{j} … T̃_{n−1} ⊗ 1 for 1 ≤ j ≤ n, 0 ≤ a < d, b ∈ ℤ_{2}. But Φ maps these elements to X̃_{j}^{a}C̃_{j}^{b}T̃_{j} … T̃_{n−1}T̃_{n} which, again using (i), form a basis for H_{n}^{f}T̃_{n}H_{n}^{f} as a free right H_{n}^{f}- module. This shows that H_{n}^{f}T̃_{n}H_{n}^{f} ≃ H_{n}^{f} ⊗_{H_{n−1}^{f}} H_{n}^{f}. Now the remaining parts of (ii), (iii) are obvious consequences of (i).

We have now decomposed H_{n+1}^{f} as an (H_{n}^{f}, H_{n}^{f})-bimodule. So the same argument as for Theorem 2.8 easily gives:

Theorem 3.9. Let M be an H_{n}^{f}-module. Then, there is a natural isomorphism

res_{H_{n}^{f}}^{H_{n+1}^{f}}ind_{H_{n}^{f}}^{H_{n+1}^{f}}M ≃ (M ⊕ ΠM)^{⊕d} ⊕ ind_{H_{n−1}^{f}}^{H_{n}^{f}}res_{H_{n−1}^{f}}^{H_{n}^{f}}M

of H_{n}^{f}-modules.

§3-d. Duality. We wish next to prove that the induction functor ind_{H_{n}^{f}}^{H_{n+1}^{f}} commutes with the τ-duality. We need a little preliminary work.

Lemma 3.10. For 1 ≤ i ≤ n and a ≥ 0,

(T̃_{n} + ξC̃_{n}C̃_{n+1}) … (T̃_{i} + ξC̃_{i}C̃_{i+1})X̃_{i}^{a}T̃_{i} … T̃_{n} = X̃_{n+1}^{a} + (∗)  (3.7)

where (∗) is a term lying in H_{n}^{f}T̃_{n}H_{n}^{f} + ∑_{k=1}^{a−1} (X̃_{n+1}^{k}H_{n}^{f} + X̃_{n+1}^{k−1}C̃_{n+1}H_{n}^{f}), and

(T̃_{n} + ξC̃_{n}C̃_{n+1}) … (T̃_{i} + ξC̃_{i}C̃_{i+1})X̃_{i}^{a}C̃_{i}T̃_{i} … T̃_{n} = X̃_{n+1}^{a}C̃_{n+1} + (∗∗)  (3.8)

where (∗∗) is a term lying in H_{n}^{f}T̃_{n}H_{n}^{f} + ∑_{k=1}^{a} (X̃_{n+1}^{k}H_{n}^{f} + X̃_{n+1}^{k−1}C̃_{n+1}H_{n}^{f}).

Proof. We prove (3.7) and (3.8) simultaneously by induction on n = i, i + 1, . . .. In case n = i, they follow from a calculation involving (2.23) or (2.24) respectively, together with Lemma 3.7 to commute X̃_{n+1}^{a} and X̃_{n+1}^{a}C̃_{n+1} past T̃_{n}. The induction step is similar, noting that both (T̃_{n+1} + ξC̃_{n+1}C̃_{n+2}) and T̃_{n+1} centralize H_{n}^{f}.

Lemma 3.11. For any s ∈ H_{n}^{f},

X̃_{n+1}^{d}s = −a_{0}s + (a term lying in H_{n}^{f}T̃_{n}H_{n}^{f} + ∑_{k=1}^{d−1} (X̃_{n+1}^{k}H_{n}^{f} + X̃_{n+1}^{k−1}C̃_{n+1}H_{n}^{f})) .

Proof. Recall the polynomial f ∈ F [X_{1}] from (3.1). Its image f̃ ∈ H_{n}^{f} is zero. Hence,

(T̃_{n} + ξC̃_{n}C̃_{n+1}) … (T̃_{1} + ξC̃_{1}C̃_{2})f̃T̃_{1} … T̃_{n} = 0.

But a calculation using (3.7) shows that the left hand side equals X̃_{n+1}^{d} + a_{0} modulo terms of the given form. The lemma follows easily on multiplying on the right by s.

Lemma 3.12. There exists an even (H_{n}^{f}, H_{n}^{f})-bimodule homomorphism θ : H_{n+1}^{f} → H_{n}^{f} such that ker θ contains no non-zero left ideals of H_{n+1}^{f}. Proof. By Lemma 3.8(ii), we know that

H_{n+1}^{f} = H_{n}^{f} ⊕ C̃_{n+1}H_{n}^{f} ⊕ ⨁_{a=1}^{d−1} (X̃_{n+1}^{a}H_{n}^{f} ⊕ X̃_{n+1}^{a}C̃_{n+1}H_{n}^{f}) ⊕ H_{n}^{f}T̃_{n}H_{n}^{f}

as an (H_{n}^{f}, H_{n}^{f})-bimodule. Let θ : H_{n+1}^{f} → H_{n}^{f} be the projection onto the first summand of this bimodule decomposition. We just need to show that if y ∈ H_{n+1}^{f} has the property that θ(hy) = 0 for all h ∈ H_{n+1}^{f}, then y = 0. Using Lemma 3.8(i), we may write

y = ∑_{a=0}^{d−1} (X̃_{n+1}^{a}s_{a} + X̃_{n+1}^{a}C̃_{n+1}t_{a}) + ∑_{a=0}^{d−1} ∑_{j=1}^{n} (X̃_{j}^{a}T̃_{j} … T̃_{n}u_{a,j} + X̃_{j}^{a}C̃_{j}T̃_{j} … T̃_{n}v_{a,j})

for s_{a}, t_{a}, u_{a,j}, v_{a,j} ∈ H_{n}^{f}. Consider θ(X̃_{n+1}^{d−1}C̃_{n+1}y). An application of Lemma 3.7 reveals that this equals t_{d−1}, hence t_{d−1} = 0. Next consider θ(X̃_{n+1}y). Using Lemma 3.11 as well as Lemma 3.7 this time, we get that s_{d−1} = 0. Now consider similarly θ(X̃_{n+1}^{d−2}C̃_{n+1}y), θ(X̃_{n+1}^{2}y), θ(X̃_{n+1}^{d−3}C̃_{n+1}y), θ(X̃_{n+1}^{3}y), … in turn to deduce t_{d−2} = s_{d−2} = t_{d−3} = s_{d−3} = ⋯ = 0.

We have now reduced to the case that

y = ∑_{a=0}^{d−1} ∑_{j=1}^{n} (X̃_{j}^{a}T̃_{j} … T̃_{n}u_{a,j} + X̃_{j}^{a}C̃_{j}T̃_{j} … T̃_{n}v_{a,j}) .

Now consider y′ := (T̃_{n} + ξC̃_{n}C̃_{n+1})y. Note

(T̃_{n} + ξC̃_{n}C̃_{n+1})T̃_{n−1}T̃_{n} = T̃_{n−1}T̃_{n}(T̃_{n−1} + ξC̃_{n−1}C̃_{n})

so the terms of y with j < n yield terms of y′ which lie in H_{n}^{f}T̃_{n}H_{n}^{f}. Hence, by Lemma 3.10 too,

y′ = X̃_{n+1}^{d−1}C̃_{n+1}v_{d−1,n} + (∗)

where (∗) is a term lying in H_{n}^{f}T̃_{n}H_{n}^{f} + ∑_{k=1}^{d−1} (X̃_{n+1}^{k}H_{n}^{f} + X̃_{n+1}^{k−1}C̃_{n+1}H_{n}^{f}). Now multiplying y′

by X̃_{n+1}^{d−1}C̃_{n+1} and applying θ, as in the previous paragraph, gives that v_{d−1,n} = 0. Hence in fact, by Lemma 3.10 once more, we have that

y′ = X̃_{n+1}^{d−1}u_{d−1,n} + (∗)

and now one gets u_{d−1,n} = 0 on multiplying by X̃_{n+1} and applying θ, again as in the previous paragraph. Continuing in this way gives that all u_{a,n} = v_{a,n} = 0.

Now repeat the argument in the previous paragraph again, this time considering y′ := (T̃_{n} + ξC̃_{n}C̃_{n+1})(T̃_{n−1} + ξC̃_{n−1}C̃_{n})y, to get that all u_{a,n−1} = v_{a,n−1} = 0. Continuing in this way eventually gives the desired conclusion: y = 0.

Now we are ready to prove the main result of the subsection:

Theorem 3.13. There is a natural isomorphism H_{n+1}^{f} ⊗_{H_{n}^{f}} M ≃ Hom_{H_{n}^{f}}(H_{n+1}^{f}, M) for all

H_{n}^{f}-modules M.

Proof. We show that there is an even isomorphism φ : H_{n+1}^{f} → Hom_{H_{n}^{f}}(H_{n+1}^{f}, H_{n}^{f}) of

(H_{n+1}^{f}, H_{n}^{f})-bimodules. The lemma then follows on applying the functor ? ⊗_{H_{n}^{f}} M: one obtains natural isomorphisms

H_{n+1}^{f} ⊗_{H_{n}^{f}} M ⟶^{φ⊗id} Hom_{H_{n}^{f}}(H_{n+1}^{f}, H_{n}^{f}) ⊗_{H_{n}^{f}} M ≃ Hom_{H_{n}^{f}}(H_{n+1}^{f}, M).

Note the existence of the second isomorphism here uses the fact that H_{n+1}^{f} is a projective left H_{n}^{f}-module, see [AF, 20.10].

To construct φ, let θ be as in Lemma 3.12, and define φ(h) to be the map hθ, for each h ∈ H_{n+1}^{f}. One easily checks that φ : H_{n+1}^{f} → Hom_{H_{n}^{f}}(H_{n+1}^{f}, H_{n}^{f}) is then a well-defined

homomorphism of (H_{n+1}^{f}, H_{n}^{f})-bimodules. To see that it is an isomorphism, it suffices by dimensions to check it is injective. Suppose φ(h) = 0 for some h ∈ H_{n+1}^{f}. Then for every x ∈ H_{n+1}^{f}, θ(xh) = 0, i.e. the left ideal H_{n+1}^{f}h is contained in ker θ. So Lemma 3.12 implies h = 0.

Corollary 3.14. H_{n}^{f} is a Frobenius superalgebra, i.e. there is an even isomorphism of left H_{n}^{f}-modules H_{n}^{f} ≃ Hom_{F} (H_{n}^{f}, F) between the left regular module and the F -linear dual of the right regular module. Proof. Proceed by induction on n. For the induction step,

H_{n}^{f} ≃ H_{n}^{f} ⊗_{H_{n−1}^{f}} H_{n−1}^{f} ≃ H_{n}^{f} ⊗_{H_{n−1}^{f}} Hom_{F} (H_{n−1}^{f}, F)
≃ Hom_{H_{n−1}^{f}}(H_{n}^{f}, Hom_{F} (H_{n−1}^{f}, F)) ≃ Hom_{F} (H_{n−1}^{f} ⊗_{H_{n−1}^{f}} H_{n}^{f}, F)
≃ Hom_{F} (H_{n}^{f}, F),

applying Theorem 3.13 and adjointness of tensor and Hom.

For the next corollary, recall the duality induced by τ (2.32) on finite dimensional H_{n}- modules. Since τ leaves the two-sided ideal I_{f} invariant, it induces a duality also denoted τ on finite dimensional H_{n}^{f}-modules.

Corollary 3.15. The exact functor ind_{H_{n}^{f}}^{H_{n+1}^{f}} is both left and right adjoint to res_{H_{n}^{f}}^{H_{n+1}^{f}}. Moreover, it commutes with duality in the sense that there is a natural isomorphism

ind_{H_{n}^{f}}^{H_{n+1}^{f}}(M^{τ}) ≃ (ind_{H_{n}^{f}}^{H_{n+1}^{f}}M)^{τ}

for all finite dimensional H_{n}^{f}-modules M.

Proof. The fact that ind_{H_{n}^{f}}^{H_{n+1}^{f}} = H_{n+1}^{f}⊗_{H_{n}^{f}}? is right adjoint to res_{H_{n}^{f}}^{H_{n+1}^{f}} is immediate from

Theorem 3.13, since Hom_{H_{n}^{f}}(H_{n+1}^{f},?) is right adjoint to restriction by adjointness of tensor and Hom. But on finite dimensional modules, a standard check using (2.2) shows that the

functor τ ∘ ind_{H_{n}^{f}}^{H_{n+1}^{f}} ∘ τ is also right adjoint to restriction. Now the remaining part of the corollary follows by uniqueness of adjoint functors.

§3-e. Modifications in the degenerate case. In the degenerate case, H_{n}^{f} becomes the cyclotomic Sergeev superalgebra defined for f ∈ F [x_{1}] ⊂ H_{n} a polynomial of the form

x_{1}^{d} + a_{d−2}x_{1}^{d−2} + a_{d−4}x_{1}^{d−4} + …,

i.e. the powers of x_{1} appearing are either all even or all odd and the leading coefficient is 1. By definition, H_{n}^{f} = H_{n}/I_{f} where I_{f} is the two-sided ideal generated by f. The basis theorem says that H_{n}^{f} has basis given by the images of

{x^{α}c^{β}w | α ∈ ℤ_{≥0}^{n} with 0 ≤ α_{i} < d, β ∈ ℤ_{2}^{n}, w ∈ S_{n}},

where I_{f} is the two-sided ideal of H_{n} generated by f.

The proof is entirely similar to that of Theorem 3.6; actually in this case it is much more straightforward. To give a little more detail, one defines the element f_{i} = s_{i−1} … s_{1}fs_{1} … s_{i−1} for each i = 1, …, n then defines f_{Z} as before for Z ⊆ {1, …, n}. By (2.37),

f_{i} = x_{i}^{d} + (a linear combination of terms lying in P_{i−1}x_{i}^{e}H_{i}^{fin} for 0 ≤ e < d).

Given this one easily proves as in Lemma 3.2 that the

{X^{α}f_{Z} | Z ⊆ {1, …, n}, α ∈ ℤ_{≥0}^{n} with 0 ≤ α_{i} < d if i ∉ Z}

form a basis for H_{n} viewed as a right H_{n}^{fin}-module. Moreover, arguing as for Lemma 3.5 the

{X^{α}f_{Z} | ∅ ≠ Z ⊆ {1, …, n}, α ∈ ℤ_{≥0}^{n} with 0 ≤ α_{i} < d if i ∉ Z}

form a basis for I_{f} as a right H_{n}^{fin}-module. Finally, the proof is completed as in Theorem 3.6.

Theorem 3.9 goes through without significant alteration. Note a suitable decomposition of H_{n+1}^{f} as an (H_{n}^{f}, H_{n}^{f})-bimodule is

H_{n+1}^{f} = ⨁_{0≤a<d,b∈ℤ_{2}} x_{n+1}^{a}c_{n+1}^{b}H_{n}^{f} ⊕ H_{n}^{f}s_{n}H_{n}^{f}.  (3.9)

To prove that induction commutes with duality, i.e. Theorem 3.13, there is a slight twist in proving the analogue of Lemma 3.12: the map θ should be taken to be the projection

θ : H_{n+1}^{f} → x_{n+1}^{d−1}H_{n}^{f} ≃ H_{n}^{f}

along the direct sum decomposition (3.9).

4. The category of integral representations

§4-a. Affine Kac-Moody algebra. Now we introduce some standard Lie theoretic notation. Let us treat the case h ≠ ∞ first, when we let ℓ = (h − 1)/2 and 𝔤 denote the twisted affine Kac-Moody algebra of type A_{2ℓ}^{(2)} (over ℂ), see [Kc, ch. 4, table Aff 2]. In particular we label the Dynkin diagram by the index set I = {0, 1, …, ℓ} as follows:

[figure: <; <; <; 0; 1; 2; ℓ−2; ℓ−1; ℓ; 0; 1; if ℓ = 1.; if ℓ ≥ 2, and]

The weight lattice is denoted P, the simple roots are {α_{i} | i ∈ I} ⊂ P and the corresponding simple coroots are {h_{i} | i ∈ I} ⊂ P^{∗}. The Cartan matrix (⟨h_{i}, α_{j}⟩)_{0≤i,j≤ℓ} is

(\begin{matrix}2 & −2 & 0 & ⋯ & 0 & 0 & 0 \\ −1 & 2 & −1 & ⋯ & 0 & 0 & 0 \\ 0 & −1 & 2 & ⋯ & 0 & 0 & 0 \\ . & . & . \\ 0 & 0 & 0 & … & 2 & −1 & 0 \\ 0 & 0 & 0 & … & −1 & 2 & −2 \\ 0 & 0 & 0 & … & 0 & −1 & 2 \end{matrix}) if ℓ ≥ 2, and (\begin{matrix}2 & −4 \\ −1 & 2 \end{matrix}) if ℓ = 1.

Let {Λ_{i} | i ∈ I} ⊂ P denote fundamental dominant weights, so that ⟨h_{i}, Λ_{j}⟩ = δ_{i,j}, and let P_{+} ⊂ P denote the set of all dominant integral weights. Set

c = h_{0} + ∑_{i=1}^{ℓ} 2h_{i},  δ = ∑_{i=0}^{ℓ−1} 2α_{i} + α_{ℓ}.  (4.1)

Then the Λ_{0}, …, Λ_{ℓ}, δ form a ℤ-basis for P, and ⟨c, α_{i}⟩ = ⟨h_{i}, δ⟩ = 0 for all i ∈ I.

In the case h = ∞, we make the following changes to these definitions. First, we let ℓ = ∞, and 𝔤 denotes the Kac-Moody algebra of type B_{∞}, see [Kc, §7.11]. So I = {0, 1, 2, … }, corresponding to the nodes of the Dynkin diagram

[figure: <; 0; 1; 2]

Note certain notions, for example the element c from (4.1), only make sense if one passes to the completed algebra b_{∞}, see [Kc, §7.12], though the intended meaning whenever we make use of them should be obvious regardless.

Now, for either h < ∞ or h = ∞, we let U_{ℚ} denote the ℚ-subalgebra of the universal enveloping algebra of 𝔤 generated by the Chevalley generators e_{i}, f_{i}, h_{i} (i ∈ I). Recall these are subject only to the relations

[h_{i}, h_{j}] = 0,  [e_{i}, f_{j}] = δ_{i,j}h_{i},  (4.2)
[h_{i}, e_{j}] = ⟨h_{i}, α_{j}⟩e_{j},  [h_{i}, f_{j}] = −⟨h_{i}, α_{j}⟩f_{j},  (4.3)
(ad e_{i})^{1−⟨h_{i},α_{k}⟩}e_{k} = 0,  (ad f_{i})^{1−⟨h_{i},α_{k}⟩}f_{k} = 0  (4.4)

for all i, j, k ∈ I with i ≠ k. We let U_{ℤ} denote the ℤ-form of U_{ℚ} generated by the divided powers e_{i}^{(n)} = e_{i}^{n}/n! and f_{i}^{(n)} = f_{i}^{n}/n!. Then, U_{ℤ} has the usual triangular decomposition

U_{ℤ} = U_{ℤ}^{−}U_{ℤ}^{0}U_{ℤ}^{+}.

We are particularly concerned here with the plus part U_{ℤ}^{+}, generated by all e_{i}^{(n)}. It is a graded Hopf algebra over ℤ via the principal grading deg(e_{i}^{(n)}) = n for all i ∈ I, n ≥ 0.

§4-b. Cyclotomic Hecke-Clifford superalgebras revisited. Given i ∈ I, define

q(i) := 2\frac{q^{2i+1} + q^{−2i−1}}{q + q^{−1}} ∈ F.  (4.5)

Note in particular that q(0) = 2. For λ ∈ P_{+}, let I_{λ} denote the two-sided ideal of H_{n} generated by the element

(X_{1} − 1)^{⟨h_{0},λ⟩} ∏_{i=1}^{ℓ}(X_{1} + X_{1}^{−1} − q(i))^{⟨h_{i},λ⟩}  (4.6)

Up to a power of the unit X_{1}, this is an element of the form (3.1), so the quotient superalgebra

H_{n}^{λ} := H_{n}/I_{λ}

is a special case of the cyclotomic Hecke-Clifford superalgebras introduced in the previous section. This quotient of H_{n} defined for λ ∈ P_{+} should not be confused with the parabolic subalgebra H_{μ} defined earlier for μ a composition of n.

Theorem 3.6 immediately gives the following basis theorem for H_{n}^{λ}:

Theorem 4.1. For any λ ∈ P_{+}, the canonical images of the elements

{X^{α}C^{β}T_{w} | α ∈ ℤ^{n} with 0 ≤ α_{1}, …, α_{n} < ⟨c, λ⟩, β ∈ ℤ_{2}^{n}, w ∈ S_{n}}

form a basis for H_{n}^{λ}. In particular, dim H_{n}^{λ} = (2⟨c, λ⟩)^{n}(n!). Remark 4.2. In the special case λ = Λ_{0} is the first fundamental dominant weight, the cyclotomic Hecke-Clifford superalgebra H_{n}^{λ} can be identified with the superalgebra H_{n}^{fin}. This follows easily from Theorem 4.1: since ⟨c, Λ_{0}⟩ = 1, H_{n}^{Λ_{0}} has basis give by the images of the elements {C^{β}T_{w} | β ∈ ℤ_{2}^{n}, w ∈ S_{n}} just as H_{n}^{fin}, and the multiplications are the same by construction. See also [JN, Prop. 3.5].

Introduce the functors

pr^{λ} : H_{n}-mod → H_{n}^{λ}-mod,  infl^{λ} : H_{n}^{λ}-mod → H_{n}-mod .  (4.7)

Here, infl^{λ} is simply inflation along the canonical epimorphism H_{n} → H_{n}^{λ}, while on a module M, pr^{λ}M = M/I_{λ}M with the induced action of H_{n}^{λ}. The functor infl^{λ} is right adjoint to pr^{λ}, i.e. there is a functorial isomorphism

Hom_{H_{n}^{λ}}(pr^{λ}M, N) ≃ Hom_{H_{n}}(M, infl^{λ}N).  (4.8)

Note we will generally be sloppy and omit the functor infl^{λ} in our notation. In other words, we generally identify H_{n}^{λ}-mod with the full subcategory of H_{n}-mod consisting of all modules M with I_{λ}M = 0.

§4-c. Elements Φ̃_{j}. We will need certain elements of H_{n} defined originally by Jones and Nazarov. Given 1 ≤ j < n, define

z_{j} := X_{j} + X_{j}^{−1} − X_{j+1} − X_{j+1}^{−1} = X_{j}^{−1}(X_{j}X_{j+1} − 1)(X_{j}X_{j+1}^{−1} − 1),  (4.9)
Φ̃_{j} := z_{j}^{2}T_{j} + ξ \frac{z_{j}^{2}}{X_{j}X_{j+1}^{−1} − 1} − ξ \frac{z_{j}^{2}}{X_{j}X_{j+1} − 1}C_{j}C_{j+1}.  (4.10)

Then Φ̃_{j} is equal to z_{j}^{2}Φ_{j} where Φ_{j} is the element defined by Jones and Nazarov in [JN, (3.6)]. Note Φ̃_{j} really does make sense as an element of H_{n}, unlike Φ_{j} which belongs to a certain localization. An easy calculation as in [JN, (3.7)] gives that

Φ̃_{j}X_{j}^{±1} = X_{j+1}^{±1}Φ̃_{j},  Φ̃_{j}X_{j+1}^{±1} = X_{j}^{±1}Φ̃_{j},  Φ̃_{j}X_{k}^{±1} = X_{k}^{±1}Φ̃_{j},  (4.11)
Φ̃_{j}C_{j} = C_{j+1}Φ̃_{j},  Φ̃_{j}C_{j+1} = C_{j}Φ̃_{j},  Φ̃_{j}C_{k} = C_{k}Φ̃_{j}  (4.12)

for k ≠ j, j + 1. Moreover, [JN, Prop. 3.1] implies that

Φ̃_{j}^{2} = z_{j}^{2}(X_{j}^{−2}(X_{j}X_{j+1} − 1)^{2}(X_{j}X_{j+1}^{−1} − 1)^{2} − ξ^{2}X_{j}^{−1}X_{j+1}^{−1}(X_{j}X_{j+1} − 1)^{2}

− ξ^{2}X_{j}^{−1}X_{j+1}(X_{j}X_{j+1}^{−1} − 1)^{2}). (4.13)

In order to make use of this, we need the following technical lemma:

Lemma 4.3. Suppose a, b ∈ F^{×} with b + b^{−1} = q(i) for some i ∈ I. If

a^{−2}(ab − 1)^{2}(ab^{−1} − 1)^{2}(a^{−2}(ab − 1)^{2}(ab^{−1} − 1)^{2} − ξ^{2}a^{−1}b^{−1}(ab − 1)^{2}

− ξ^{2}a^{−1}b(ab^{−1} − 1)^{2}) = 0.

then a + a^{−1} = q(j) for j ∈ I with |i − j| ≤ 1.

Proof. Follow [JN, (4.1)–(4.4)].

A more lengthy calculation also as in [JN, Prop. 3.1] shows that the elements Φ̃_{j} satisfy the braid relations, i.e.

Φ̃_{i}Φ̃_{j} = Φ̃_{j}Φ̃_{i},  Φ̃_{i}Φ̃_{i+1}Φ̃_{i} = Φ̃_{i+1}Φ̃_{i}Φ̃_{i+1}.  (4.14)

for all admissible i, j with |i − j| > 1. This means that for any w ∈ S_{n}, we obtain welldefined elements Φ̃_{w} ∈ H_{n}, namely, Φ̃_{w} := Φ̃_{i_{1}} … Φ̃_{i_{m}} where w = s_{i_{1}} … s_{i_{m}} is any reduced expression for w. According to (4.11),(4.12), these elements have the property that

Φ̃_{w}X_{i}^{±1} = X_{wi}^{±1}Φ̃_{w},  Φ̃_{w}C_{i} = C_{wi}Φ̃_{w},  (4.15)

for all w ∈ S_{n}, 1 ≤ i ≤ n. Note we will not make essential use of the fact that the Φ̃_{i} satisfy the braid relations in what follows.

§4-d. Integral representations. Now call an A_{n}-module M integral if it is finite dimensional and moreover all eigenvalues of X_{1} +X_{1}^{−1}, …, X_{n} +X_{n}^{−1} on M are of the form q(i) for i ∈ I, see (4.5). Call an H_{n}-module, or more generally an H_{μ}-module for μ a composition of n, integral if it is integral on restriction to A_{n}. In what follows we will restrict our attention to these modules, and write Rep_{I} H_{n} (resp. Rep_{I} A_{n}, Rep_{I} H_{μ}) for the full subcategory of H_{n}-mod (resp. A_{n}-mod, H_{μ}-mod) consisting of all integral modules.

Lemma 4.4. Let M be a finite dimensional H_{n}-module, and 1 ≤ j ≤ n. Assume that the eigenvalues of X_{j} + X_{j}^{−1} on M are of the form q(i), i ∈ I. Then the same is true for the eigenvalues of all other X_{k} + X_{k}^{−1}, k = 1, 2, …, n. Proof. It suffices to show that the eigenvalues of X_{j} + X_{j}^{−1} are of the form q(i) if and only if the eigenvalues of X_{j+1} + X_{j+1}^{−1} are of the same form, for 1 ≤ j < n. Actually, by an argument involving conjugation with the automorphism σ, it suffices just to prove the ‘if’ part. So assume that all eigenvalues of X_{j+1} + X_{j+1}^{−1} on M are of the form q(i) for various i ∈ I. Let a ≠ 0 be an eigenvalue for the action of X_{j} on M. We have to prove that a + a^{−1} is also of the form q(i). Since X_{j} and X_{j+1} commute, we can pick v lying in the a-eigenspace of X_{j} so that v is also an eigenvector for X_{j+1}, of eigenvalue b say. By assumption, b + b^{−1} = q(i) for some i ∈ I. Now let Φ̃_{j} be the element (4.10). By (4.11), (X_{j+1} + X_{j+1}^{−1})Φ̃_{j} = Φ̃_{j}(X_{j} + X_{j}^{−1}). So if Φ̃_{j}v ≠ 0, we get that

(a + a^{−1})Φ̃_{j}v = Φ̃_{j}(X_{j} + X_{j}^{−1})v = (X_{j+1} + X_{j+1}^{−1})Φ̃_{j}v

so that a + a^{−1} = q(i′) for some i′ ∈ I by assumption. Else, Φ̃_{j}v = 0 so Φ̃_{j}^{2}v = 0. So applying (4.13) and Lemma 4.3, we again get that a + a^{−1} = q(i′) for some i′ ∈ I.

Corollary 4.5. Let M be a finite dimensional H_{n}-module. Then M is integral if and only if I_{λ}M = 0 for some λ ∈ P_{+}. Proof. If I_{λ}M = 0, then the eigenvalues of X_{1} + X_{1}^{−1} on M are of the form q(i) for i ∈ I, by definition of I_{λ}. Hence M is integral in view of Lemma 4.4. Conversely, suppose that M is integral. Then the minimal polynomial of X_{1} + X_{1}^{−1} on M is of the form ∏_{i∈I}(t − q(i))^{λ_{i}} for some λ_{i} ≥ 0. So if we set λ = 2λ_{0}Λ_{0} + λ_{1}Λ_{1} + ⋯ + λ_{ℓ}Λ_{ℓ} ∈ P_{+}, we certainly have that the element (4.6) acts as zero on M.

Recall from §2-c that Rep H_{n}^{λ} denotes the category of all finite dimensional H_{n}^{λ}-modules. Corollary 4.5 implies that the functors pr^{λ} and infl^{λ} from (4.7) restrict to a well-defined adjoint pair of functors at the level of integral representations:

pr^{λ} : Rep_{I} H_{n} → Rep H_{n}^{λ},  infl^{λ} : Rep H_{n}^{λ} → Rep_{I} H_{n}.  (4.16)

Let us also check at this point that induction from a parabolic subalgebra of H_{n} preserves integral representations, the analogous fact for restriction being obvious.

Lemma 4.6. Let μ be a composition of n and M be an integral H_{μ}-module. Then, ind_{μ}^{n}M is an integral H_{n}-module. Proof. By Theorem 2.2, ind_{μ}^{n}M is spanned by elements T_{w} ⊗ m for m ∈ M, in particular it is finite dimensional. Let

Y_{j} = ∏_{i∈I}(X_{j} + X_{j}^{−1} − q(i)).

By Corollary 4.5, it suffices to show that Y_{1}^{N} annihilates ind_{λ}^{n}M for sufficiently large N. Consider Y_{1}^{N}T_{w} ⊗ m for w ∈ S_{n}, m ∈ M. We may write T_{w} = T_{u}T_{1} … T_{k} for u ∈ S_{2...n} ≅ S_{n−1} and 0 ≤ k < n. Then, Y_{1}^{N} commutes with T_{u}, so we just need to consider Y_{1}^{N}T_{1} … T_{k}⊗ m. Now using the commutation relations, one checks that Y_{1}^{N}T_{1} … T_{k} ⊗m can be rewritten as an H_{n}-linear combination of elements of the form 1⊗Y_{j}^{N′}m for j ∈ I and N −k ≤ N′ ≤ N. Since M is integral by assumption, we can choose N sufficiently large so that each such term is zero.

It follows that the functors ind_{μ}^{n}, res_{μ}^{n} restrict to well-defined functors

ind_{μ}^{n} : Rep_{I} H_{μ} → Rep_{I} H_{n},  res_{μ}^{n} : Rep_{I} H_{n} → Rep_{I} H_{μ}  (4.17)

on integral representations. Similar remarks apply to more general induction and restriction between nested parabolic subalgebras of H_{n}.

§4-e. Modules over A_{n}. Let i ∈ I and define

b_{±}(i) = \frac{q(i)}{2} ± \sqrt{\frac{q(i)^{2}}{4} − 1},  (4.18)

i.e. the roots of the equation x + x^{−1} = q(i). Let L(i) denote the vector superspace on basis w, w′, where w is even and w′ is odd, made into an A_{1}-module so that

C_{1}w = w′, C_{1}w′ = w, X_{1}^{±1}w = b_{±}(i)w, X_{1}^{±1}w′ = b_{∓}(i)w′.

One easily checks:

Lemma 4.7. For each i ∈ I, L(i) is an irreducible A_{1}-module, of type M if i ≠ 0 and of type Q if i = 0. Moreover, the modules {L(i) | i ∈ I} form a complete set of pairwise non-isomorphic irreducibles in Rep_{I} A_{1}.

Recall that A_{n} ≅ A_{1}⊗⋯⊗A_{1} (n times) as superalgebras. Hence, for i = (i_{1}, …, i_{n}) ∈ I^{n}, we can consider the irreducible A_{n}-module L(i_{1})⊛⋯⊛L(i_{n}). By Lemma 4.7 and the general theory of outer tensor products §2-b, one obtains:

Lemma 4.8. The A_{n}-modules {L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}) | i ∈ I^{n}} form a complete set of pairwise non-isomorphic irreducible A_{n}-modules. Moreover, let γ_{0} denote the number of j = 1, …, n such that i_{j} = 0. Then, L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}) is of type M if γ_{0} is even and type Q if γ_{0} is odd. Finally, dim L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}) = 2^{n−⌊γ_{0}/2⌋}.

Now let M be any module in Rep_{I} A_{n}. For any i ∈ I^{n}, let M[i] be the largest submodule of M all of whose composition factors are isomorphic to L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}). Alternatively, since each X_{k} +X_{k}^{−1} acts on L(i_{1})⊛⋯⊛L(i_{n}) by the scalar q(i_{k}) and all the scalars q(i) for i ∈ I are distinct, we can describe M[i] as the simultaneous generalized eigenspace for the commuting operators X_{1} +X_{1}^{−1}, …, X_{n} +X_{n}^{−1} corresponding to eigenvalues q(i_{1}), …, q(i_{n}), respectively. Hence:

Lemma 4.9. For any M ∈ Rep_{I} A_{n}, M = ⨁_{i∈I^{n}} M[i] as an A_{n}-module.

We write K(Rep_{I} A_{n}), K(Rep_{I} H_{n}), … for the Grothendieck groups of the categories Rep_{I} A_{n}, Rep_{I} H_{n}, …, defined as in §2-c. Note for an integral A_{n}-module M, knowledge of the dimensions of the spaces M[i] for all i is equivalent to knowing the coefficients a_{i} when [M] ∈ K(Rep_{I} A_{n}) is expanded as

[M] = ∑_{i∈I^{n}} a_{i}[L(i_{1}) ⊛ ⋯ ⊛ L(i_{n})]

in terms of the basis {[L(i_{1}) ⊛ ⋯ ⊛ L(i_{n})] | i ∈ I^{n}}.

Now suppose instead that M is an integral H_{n}-module, so that its restriction res_{1,...,1}^{n}M to A_{n} is in Rep_{I} A_{n}. We define the formal character of M by:

ch M = [res_{1,...,1}^{n}M] ∈ K(Rep_{I} A_{n}).  (4.19)

Since the functor res_{1,...,1}^{n} is exact, ch induces a homomorphism

ch : K(Rep_{I} H_{n}) → K(Rep_{I} A_{n})

at the level of Grothendieck groups. We will later see that this map is actually injective (Theorem 5.12), justifying the terminology. Note we will occasionally consider characters of integral modules over parabolic subalgebras H_{μ} for μ a composition of n, or over the cyclotomic algebras H_{n}^{λ} for λ ∈ P_{+}. The definitions are modified in these cases in obvious ways.

Lemma 4.10. Let i = (i_{1}, …, i_{n}) ∈ I^{n}. Then

ch ind_{1,...,1}^{n}L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}) = ∑_{w∈S_{n}}[L(i_{w^{−1}1}) ⊛ ⋯ ⊛ L(i_{w^{−1}n})]

Proof. This follows from Theorem 2.8 with μ = ν = (1^{n}).

Lemma 4.11. (“Shuffle Lemma”) Let n = m+k, and let M ∈ Rep_{I} H_{m} and K ∈ Rep_{I} H_{k} be irreducible. Assume

ch M = ∑_{i∈I^{m}} a_{i}[L(i_{1}) ⊛ ⋯ ⊛ L(i_{m})], ch K = ∑_{j∈I^{k}} b_{j}[L(j_{1}) ⊛ ⋯ ⊛ L(j_{k}))].

Then

ch ind_{m,k}^{n}M ⊛ K = ∑_{i∈I^{m}} ∑_{j∈I^{k}} a_{i}b_{j}(∑_{h} L(h_{1}) ⊛ ⋯ ⊛ L(h_{n})),

where the last sum is over all h = (h_{1}, …, h_{n}) ∈ I^{n} which are obtained by shuffling i and j, i.e. there exist 1 ≤ u_{1} < ⋯ < u_{m} ≤ n such that (h_{u_{1}}, …, h_{u_{m}}) = (i_{1}, …, i_{m}), and (h_{1}, …, ĥ_{u_{1}}, …, ĥ_{u_{m}}, …, h_{n}) = (j_{1}, …, j_{k}). Proof. This follows from Theorem 2.8 with μ = (1^{n}) and ν = (m, k).

§4-f. Central characters. Recall by Theorem 2.3 that every element z of the center Z(H_{n}) of H_{n} can be written as a symmetric polynomial f(X_{1} + X_{1}^{−1}, …, X_{n} + X_{n}^{−1}) in the X_{k} + X_{k}^{−1}. Given i ∈ I^{n}, we associate the central character

χ_{i} : Z(H_{n}) → F, f(X_{1} + X_{1}^{−1}, …, X_{n} + X_{n}^{−1}) ↦ f(q(i_{1}), …, q(i_{n})).

Consider the natural left action of S_{n} on I^{n} by place permutation. We write i ∼ j if i, j lie in the same orbit. The following lemma follows immediately from the fact that the q(i) are distinct as i runs over the index set I.

Lemma 4.12. For i, j ∈ I^{n}, χ_{i} = χ_{j} if and only if i ∼ j.

Given i ∈ I^{n}, we define its weight wt(i) ∈ P by

wt(i) = ∑_{i∈I} γ_{i}α_{i}  where  γ_{i} = ♯{j = 1, …, n | i_{j} = i}.  (4.20)

So wt(i) is an element of the set Γ_{n} of non-negative integral linear combinations γ = ∑_{i∈I} γ_{i}α_{i} of the simple roots such that ∑_{i∈I} γ_{i} = n. Obviously, the S_{n}-orbit of i is uniquely determined by its weight, so we obtain a labelling of the orbits of S_{n} on I^{n} by the elements of Γ_{n}. We will also use the notation χ_{γ} for the central character χ_{i} where i is any element of I^{n} with wt(i) = γ. So χ_{i} = χ_{wt(i)}.

Now let M be an integral H_{n}-module and γ = ∑_{i∈I} γ_{i}α_{i} ∈ Γ_{n}. We let M[γ] denote the generalized eigenspace of M over Z(H_{n}) that corresponds to the central character χ_{γ}, i.e.

M[γ] = {m ∈ M | (z − χ_{γ}(z))^{k}m = 0 for all z ∈ Z(H_{n}) and k ≫ 0}.

Observe this is an H_{n}-submodule of M. Now, for any i ∈ I^{n} with wt(i) = γ, Z(H_{n}) acts on L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}) via the central character χ_{γ}. So applying Lemma 4.12, we see that

M[γ] = ⨁_{i with wt(i)=γ} M[i],

recalling the decomposition of M as an A_{n}-module from Lemma 4.9. Therefore:

Lemma 4.13. Any integral H_{n}-module M decomposes as

M = ⨁_{γ∈Γ_{n}} M[γ]

as an H_{n}-module.

Thus the {χ_{γ} |γ ∈ Γ_{n}} exhaust the possible central characters that can arise in an integral H_{n}-module, while Lemma 4.10 shows that every such central character does arise in some integral H_{n}-module.

Let us write Rep_{γ} H_{n} for the full subcategory of Rep_{I} H_{n} consisting of all modules M with M[γ] = M. Then, Lemma 4.13 implies that there is an equivalence of categories

Rep_{I} H_{n} ≅ ⨁_{γ∈Γ_{n}} Rep_{γ} H_{n}.  (4.21)

We say that Rep_{γ} H_{n} is the block of Rep_{I} H_{n} corresponding to the central character χ_{γ}. In particular, if M ≠ 0 is indecomposable then M belongs to Rep_{γ} H_{n}, i.e. M = M[γ], for a unique γ ∈ Γ_{n}.

We can extend some of these notions to integral H_{n}^{λ}-modules, for λ ∈ P_{+}. In particular, if M ∈ Rep H_{n}^{λ}, we also write M[γ] for the summand M[γ] of M defined by first viewing M as an H_{n}-module by inflation. Also write Rep_{γ} H_{n}^{λ} for the full subcategory of Rep H_{n}^{λ} consisting of the modules M with M = M[γ]. Thus we also have a decomposition

Rep H_{n}^{λ} ≅ ⨁_{γ∈Γ_{n}} Rep_{γ} H_{n}^{λ}  (4.22)

induced by (4.21). Note though that we should not yet refer to Rep_{γ} H_{n}^{λ} as a block of Rep H_{n}^{λ}: the center of H_{n}^{λ} may be larger than the image of the center of H_{n}, so we cannot yet assert that Z(H_{n}^{λ}) acts on M[γ] by a single central character. Also we no longer know precisely which γ ∈ Γ_{n} have the property that Rep_{γ} H_{n}^{λ} is non-trivial. These questions will be settled in §8-d.

§4-g. Kato’s theorem. Let i ∈ I. Define Kato’s module:

L(i^{n}) := ind_{1,...,1}^{n}L(i) ⊛ ⋯ ⊛ L(i).

By Lemma 4.10, we know immediately that ch L(i^{n}) = n![L(i) ⊛ ⋯ ⊛ L(i)], hence L(i^{n}) belongs to the block Rep_{nα_{i}} H_{n}. In particular, for each k = 1, …, n, the only eigenvalue of the element X_{k} + X_{k}^{−1} on L(i^{n}) is q(i). In addition, if i = 0, then the only eigenvalue of each X_{k} is 1.

Lemma 4.14. Let n ≥ 2, 1 ≤ j < n, i ∈ I − {0}, and v ∈ L(i) ⊠ ⋯ ⊠ L(i) (n copies). Then X_{j}^{−1}(1 − C_{j}C_{j+1})v ≠ (1 − C_{j}C_{j+1})X_{j+1}v. Proof. The elements of A_{n} which are involved in the inequality act only on the positions j and j + 1 in the tensor product. So we may assume that n = 2 and j = 1. Let

v = aw ⊗ w + bw ⊗ w′ + cw′ ⊗ w + dw′ ⊗ w′

for a, b, c, d ∈ F. Then

X_{1}^{−1}(1 − C_{1}C_{2})v = (b_{−}(i)a + b_{−}(i)d)w ⊗ w + (b_{−}(i)b + b_{−}(i)c)w ⊗ w′

+ (b_{+}(i)c − b_{+}(i)b)w′ ⊗ w + (b_{+}(i)d − b_{+}(i)a)w′ ⊗ w′,

(1 − C_{1}C_{2})X_{2}v = (b_{+}(i)a + b_{−}(i)d)w ⊗ w + (b_{−}(i)b + b_{+}(i)c)w ⊗ w′

+ (b_{+}(i)c − b_{−}(i)b)w′ ⊗ w + (b_{−}(i)d − b_{+}(i)a)w′ ⊗ w′.

Now the lemma follows from the inequality b_{−}(i) ≠ b_{+}(i) for i ≠ 0.

Lemma 4.15. Let i ∈ I. Set L = L(i) ⊛ ⋯ ⊛ L(i), so L(i^{n}) = H_{n} ⊗_{A_{n}} L.

(i) If i ≠ 0, the common q(i)-eigenspace of the operators X_{1} + X_{1}^{−1}, …, X_{n−1} + X_{n−1}^{−1} on L(i^{n}) is precisely 1 ⊗ L, which is contained in the q(i)-eigenspace of X_{n} + X_{n}^{−1} too. Moreover, all Jordan blocks of X_{1} + X_{1}^{−1} on L(i^{n}) are of size n.

(ii) If i = 0, the common 1-eigenspace of the operators X_{1}, …, X_{n−1} on L(i^{n}) is precisely 1 ⊗ L, which is contained in the 1-eigenspace of X_{n} too. Moreover, all Jordan blocks of X_{1} on L(i^{n}) are of size n.

Proof. We prove (i), (ii) being similar. Note L(i^{n}) = ⨁_{x∈S_{n}} T_{x} ⊗L, since by Theorem 2.2 we know that H_{n} is a free right A_{n}-module on basis {T_{x} | x ∈ S_{n}}.

We first show that the eigenspace of X_{1}+X_{1}^{−1} is a sum of the subpaces of the form T_{y} ⊗L, where y ∈ S_{2...n} ≅ S_{n−1} is the subgroup of S_{n} generated by s_{2}, …, s_{n−1}. Well, any T_{x} can be written as T_{y}T_{1}T_{2} … T_{j} for some y ∈ S_{2...n} and 0 ≤ j < n. Note

(X_{j+1} + X_{j+1}^{−1} − q(i))v = 0

for any v ∈ L, by definition of L. Now the defining relations of H_{n} especially (2.21), (2.22) imply

(X_{1} + X_{1}^{−1} − q(i))T_{y}T_{1}T_{2} … T_{j} ⊗ v =

ξT_{y}T_{1} … T_{j−1} ⊗ (X_{j}^{−1}(1 − C_{j}C_{j+1}) − (1 − C_{j}C_{j+1})X_{j+1})v + (∗),

where (∗) stands for a sum of terms which belong to subspaces of the form T_{y′}T_{1} … T_{k} ⊗ L for y′ ∈ S_{2...n} and 0 ≤ k < j − 1.

Now assume that a linear combination

z := ∑_{y∈S_{2...n}} ∑_{0≤j<n} ∑_{v∈L} c_{y,j,v}T_{y}T_{1}T_{2} … T_{j} ⊗ v

is an eigenvector for X_{1} + X_{1}^{−1}. Then it must be annihilated by X_{1} + X_{1}^{−1} − q(i). Choose the maximal j for which the coefficient c_{y,j,v} is non-zero, and for this j choose the maximal (with respect to the Bruhat order) y such that c_{y,j,v} is non-zero. Then the calculation above and Lemma 4.14 show that (X_{1} + X_{1}^{−1} − q(i))z ≠ 0 unless j = 0. This proves our claim on the eigenspace of X_{1} + X_{1}^{−1}.

Now apply the same argument to see that the common eigenspace of X_{1} + X_{1}^{−1} and X_{2} + X_{2}^{−1} is spanned by T_{y} ⊗ L for y ∈ S_{3...n}, and so on, yielding the first claim in (i). Finally, define

V (m) := {z ∈ L(i^{n}) | (X_{1} + X_{1}^{−1} − q(i))^{m}z = 0}.

It follows by induction from the calculation above and Lemma 4.14 that

V (m) = span{T_{y}T_{1}T_{2} … T_{j} ⊗ v | y ∈ S_{2...n}, j < m, v ∈ L},

giving the second claim.

Now we are ready to prove the main theorem giving the structure of Kato’s modules.

Theorem 4.16. Let i ∈ I and μ = (μ_{1}, …, μ_{u}) be a composition of n.

(i) L(i^{n}) is irreducible of the same type as L(i) ⊛ ⋯ ⊛ L(i), i.e. type M if either i ≠ 0 or i = 0 and n is even, type Q otherwise, and it is the only irreducible module in its block.

(ii) All composition factors of res_{μ}^{n}L(i^{n}) are isomorphic to L(i^{μ_{1}}) ⊛ ⋯ ⊛ L(i^{μ_{u}}), and

soc res_{μ}^{n}L(i^{n}) is irreducible. (iii) soc res_{n−1}^{n}L(i^{n}) ≅ res_{n−1}^{n−1,1}L(i^{n−1}) ⊛ L(i).

Proof. Denote L(i) ⊛ ⋯ ⊛ L(i) by L.

(i) Let M be a non-zero H_{n}-submodule of L(i^{n}). Then, res_{1,...,1}^{n}M must contain an A_{1}- submodule N isomorphic to L. But the commuting operators X_{1} + X_{1}^{−1}, …, X_{n} + X_{n}^{−1} (or X_{1}, …, X_{n} if i = 0) act on L as scalars, giving that N is contained in their common eigenspace on L(i^{n}). But by Lemma 4.15, this implies that N = 1 ⊗ L. This shows that M contains 1 ⊗ L, but this generates the whole of L(i^{n}) over H_{n}. So M = L(i^{n}).

To see that the type of L(i^{n}) is the same as the type of L, the functor ind_{1,...,1}^{n} determines a map

End_{A_{n}}(L) → End_{H_{n}}(L(i^{n})).

We just need to see that this is an isomorphism, which we do by constructing the inverse map. Let f ∈ End_{H_{n}}(L(i^{n})). Then f leaves 1 ⊗ L invariant by Lemma 4.15, so f restricts to an A_{n}-endomorphism f̄ of L.

Finally, to see that L(i^{n}) is the only irreducible in its block, we have already observed using Lemma 4.10 that ch L(i^{n}) = n![L(i) ⊛ ⋯ ⊛ L(i)]. Hence all composition factors of res_{1,...,1}^{n}L(i^{n}) are isomorphic to L(i) ⊛ ⋯ ⊛ L(i). Now apply Frobenius reciprocity and the fact just proved that L(i^{n}) is irreducible.

(ii) That all composition factors of res_{μ}^{n}L(i^{n}) are isomorphic to L(i^{μ_{1}}) ⊛ ⋯ ⊛ L(i^{μ_{u}}) follows from the parabolic analogue of (i). To see that soc res_{μ}^{n}L(i^{n}) is simple, note that the submodule H_{μ} ⊗ L of res_{H_{μ}}L(i^{n}) is isomorphic to L(i^{μ_{1}}) ⊛ ⋯ ⊛ L(i^{μ_{l}}). This module is irreducible, and so it is contained in the socle. Conversely, let M be an irreducible H_{μ}- submodule of L(i^{n}). Then using Lemma 4.15 as in the proof of (i), we see that M must contain 1 ⊗ L, hence H_{μ} ⊗ L.

(iii) By (ii), L(i^{n}) has a unique H_{n−1,1}-submodule isomorphic to L(i^{n−1}) ⊛ L(i), namely, H_{n−1,1}⊗L. Since this is completely reducible on restriction to H_{n−1}, it follows that H_{n−1,1}⊗ L ⊆ soc res_{n−1}^{n}L(i^{n}). Conversely, take any irreducible H_{n−1}-submodule M of L(i^{n}). The common eigenspace of X_{1} +X_{1}^{−1}, …, X_{n−1} +X_{n−1}^{−1} (resp. X_{1}, …, X_{n−1} if i = 0) on M must lie in 1 ⊗ L by Lemma 4.15. Hence, M ⊆ H_{n−1,1} ⊗ L which completes the proof.

§4-h. Covering modules. Fix i ∈ I and n ≥ 1 throughout the subsection. We will construct for each m ≥ 1 an H_{n}-module L_{m}(i^{n}) with irreducible cosocle isomorphic to L(i^{n}). Let J (i^{n}) denote the annihilator in H_{n} of L(i^{n}). Introduce the quotient superalgebra

R_{m}(i^{n}) := H_{n}/J (i^{n})^{m}  (4.23)

for each m ≥ 1. One checks that J (i^{n}) contains (X_{k} + X_{k}^{−1} − q(i))^{n!} for each k = 1, …, n. It follows easily from this that each superalgebra R_{m}(i^{n}) is finite dimensional. Moreover, by Theorem 4.16, L(i^{n}) is the unique irreducible R_{m}(i^{n})-module up to isomorphism.

Let L_{m}(i^{n}) denote a projective cover of L(i^{n}) in the category R_{m}(i^{n})-mod. For convenience, we also define L_{0}(i^{n}) = R_{0}(i^{n}) = 0. Note we know the dimension of L(i^{n}) from Lemma 4.8, and moreover L(i^{n}) is of type Q if i = 0 and n is odd, type M otherwise. Using this and the general theory of finite dimensional superalgebras, one shows:

Lemma 4.17. For each m ≥ 1,

R_{m}(i^{n}) ≃ \{ \begin{matrix}(L_{m}(i^{n}) ⊕ ΠL_{m}(i^{n}))^{⊕2^{n−1}(n!)} & if i ≠ 0, \\ (L_{m}(i^{n}) ⊕ ΠL_{m}(i^{n}))^{⊕2^{(n−2)/2}(n!)} & if i = 0 and n is even, \\ L_{m}(i^{n})^{⊕2^{(n−1)/2}(n!)} & if i = 0 and n is odd, \end{matrix}

as left H_{n}-modules. Moreover, L_{m}(i^{n}) admits an odd involution if and only if i = 0 and n is odd.

There are obvious surjections

R_{1}(i^{n}) ↞ R_{2}(i^{n}) ↞ ….  (4.24)

In the case m = 1, we certainly have that L_{1}(i^{n}) ≅ L(i^{n}); let us assume by the choice of L_{1}(i^{n}) that in fact L_{1}(i^{n}) = L(i^{n}). Then we can choose the L_{m}(i^{n}) for m > 1 so that the the maps (4.24) induce even maps

L_{1}(i^{n}) ↞ L_{2}(i^{n}) ↞ ….  (4.25)

Moreover, in case i = 0 and n is odd, we can choose the odd involutions

θ_{m} : L_{m}(i^{n}) → L_{m}(i^{n})  (4.26)

given by Lemma 4.17 in such a way that they are compatible with the maps in (4.25).

The significance of the H_{n}-modules R_{m}(i^{n}) is explained by the following lemma:

Lemma 4.18. Let M be an H_{n}-module annihilated by J (i^{n})^{k} for some k. Then, there is a natural isomorphism

Hom_{H_{n}}(R_{m}(i^{n}), M) ⟶^{∼} M

for all m ≥ k. Proof. The assumption implies that M is the inflation of an R_{m}(i^{n})-module. So

Hom_{H_{n}}(R_{m}(i^{n}), M) ≃ Hom_{R_{m}(i^{n})}(R_{m}(i^{n}), M) ≃ M,

all isomorphisms being the natural ones.

Let us finally consider the most important case n = 1 in more detail. In this case, one easily checks that the ideal J (i)^{m} is generated by (X_{1} + X_{1}^{−1} − q(i))^{m} if i ≠ 0 or (X_{1} − 1)^{m} if i = 0. It follows that

dim R_{m}(i) = \{ \begin{matrix}4m & if i ≠ 0, \\ 2m & if i = 0. \end{matrix}  (4.27)

Moreover, Lemma 4.17 shows in this case that

R_{m}(i) ≃ \{ \begin{matrix}L_{m}(i) ⊕ ΠL_{m}(i) & if i ≠ 0, \\ L_{m}(i) & if i = 0. \end{matrix}  (4.28)

Hence, dim L_{m}(i) = 2m in either case. Using this, it follows easily that L_{m}(i) can be described alternatively as the vector superspace on basis w_{1}, …, w_{m}, w′_{1}, …, w′_{m}, where each w_{k} is even and each w′_{k} is odd, with H_{1}-module structure uniquely determined by

X_{1}w_{k} = b_{+}(i)w_{k} + w_{k+1}, C_{1}w_{k} = w′_{k}

for each k = 1, …, m, interpreting w_{m+1} as 0. Using this explicit description, one now checks routinely that L_{m}(i) is uniserial with m composition factors all ≃ L(i).

We can also describe the map L_{m}(i) ↞ L_{m+1}(i) from (4.25) explicitly: it is the identity on w_{1}, …, w_{m}, w′_{1}, …, w′_{m} but maps w_{m+1} and w′_{m+1} to zero. Also, the map θ_{m} from (4.26) can be chosen so that

w_{k} ↦ \sqrt{−1}w′_{k}, w′_{k} ↦ −\sqrt{−1}w_{k}

for each k = 1, …, m.

§4-i. Modifications in the degenerate case. For i ∈ I, the definition (4.5) of q(i) becomes

q(i) = i(i + 1) ∈ F  (4.29)

for each i ∈ I. For λ ∈ P_{+}, the quotient superalgebra H_{n}^{λ}, is defined to be the quotient H_{n}/I_{λ} of the affine Sergeev superalgebra H_{n} by the two-sided ideal I_{λ} generated by

x_{1}^{⟨h_{0},λ⟩} ∏_{i=1}^{ℓ}(x_{1}^{2} − q(i))^{⟨h_{i},λ⟩}.  (4.30)

The basis theorem for H_{n}^{λ} gives that H_{n}^{λ} has a basis given by the images of the elements

{x^{α}c^{β}w | α ∈ ℤ_{≥0}^{n} with 0 ≤ α_{i} < ⟨c, λ⟩, β ∈ ℤ_{2}^{n}, w ∈ S_{n}}.  (4.31)

The definition of the category Rep_{I} H_{n} of integral representations is modified in the appropriate way to ensure that the integral representations of H_{n} are precisely the inflations of finite dimensional representations of H_{n}^{λ} for λ ∈ P_{+}. To be precise, an integral A_{n}-module now means a finite dimensional A_{n}-module such that the eigenvalues of all x_{1}^{2}, …, x_{n}^{2} are of the form q(i) for i ∈ I. The appropriate analogue of Lemma 4.4, involving elements x_{j}^{2} now of course, is proved using the elements

Φ_{j} = s_{j}(x_{j}^{2} − x_{j+1}^{2}) + (x_{j} + x_{j+1}) + c_{j}c_{j+1}(x_{j} − x_{j+1})

from [N, (3.4)]. Proofs of the basic properties of Φ_{j}, analogous to those in §4-c, can be found in [N, Prop. 3.2] and at the end of [N, §4].

The module L(i) in §4-e is now defined to be the A_{1}-module on basis w, w′ with action

cw = w′,  cw′ = w,  xw = \sqrt{q(i)}w,  xw′ = −\sqrt{q(i)}w′.

The remaining definitions go through more or less unchanged: for example for an integral H_{n}-module M, M[i] is the simultaneous generalized eigenspace of the operators x_{1}^{2}, …, x_{n}^{2} corresponding to eigenvalues q(i_{1}), …, q(i_{n}) respectively.

Kato’s theorem (Theorem 4.16) is the same, as is Lemma 4.15 when X_{i} + X_{i}^{−1} is replaced by x_{i}^{2} and eigenvalue 1 for X_{i} is replaced by eigenvalue 0 for x_{i} in the usual way. Note the main technical fact needed in the proof of Lemma 4.15 is the following: for i ≠ 0, and every 0 ≠ v ∈ L(i) ⊠ ⋯ ⊠ L(i) (n copies),

(x_{j}(1 − c_{j}c_{j+1}) + (1 − c_{j}c_{j+1})x_{j+1})v ≠ 0

for each j = 1, …, n − 1.

5. Crystal operators

§5-a. Multiplicity-free socles. The arguments in this subsection are based on [GV]. Let M ∈ Rep_{I} H_{n} and i ∈ I. Define Δ_{i}M to be the generalized q(i)-eigenspace of X_{n} + X_{n}^{−1} on M. Alternatively,

Δ_{i}M = ⨁_{i∈I^{n}, i_{n}=i} M[i],

recalling the decomposition from Lemma 4.9. Note since X_{n}+X_{n}^{−1} is central in the parabolic subalgebra H_{n−1,1} of H_{n}, Δ_{i}M is invariant under this subalgebra. So in fact, Δ_{i} can be viewed as an exact functor

Δ_{i} : Rep_{I} H_{n} → Rep_{I} H_{n−1,1},  (5.1)

being defined on morphisms simply as restriction. Clearly, there is an isomorphism of functors

res_{n−1,1}^{n} ≃ Δ_{0} ⊕ Δ_{1} ⊕ ⋯ ⊕ Δ_{ℓ}.

Slightly more generally, given m ≥ 0, define

Δ_{i^{m}} : Rep_{I} H_{n} → Rep_{I} H_{n−m,m}  (5.2)

so that Δ_{i^{m}}M is the simultaneous generalized q(i)-eigenspace of of the commuting operators X_{k} + X_{k}^{−1} for k = n − m + 1, …, n. In view of Theorem 4.16(i), Δ_{i^{m}}M can also be characterized as the largest submodule of res_{n−m,m}^{n}M all of whose composition factors are of the form N ⊛ L(i^{m}) for irreducible N ∈ Rep_{I} H_{n−m}.

The definition of Δ_{i^{m}} implies functorial isomorphisms

Hom_{H_{n−m,m}}(N ⊠ L(i^{m}), Δ_{i^{m}}M) ≃ Hom_{H_{n}}(ind_{n−m,m}^{n}N ⊠ L(i^{m}), M)

for N ∈ Rep_{I} H_{n−m}, M ∈ Rep_{I} H_{n}. For irreducible N this immediately imples:

Hom_{H_{n−m,m}}(N ⊛ L(i^{m}), Δ_{i^{m}}M) ≅ Hom_{H_{n}}(ind_{n−m,m}^{n}N ⊛ L(i^{m}), M).  (5.3)

Also from definitions we get:

Lemma 5.1. Let M ∈ Rep_{I} H_{n} with ch M = ∑_{i∈I^{n}} a_{i}[L(i_{1})⊛⋯⊛L(i_{n})]. Then we have that ch Δ_{i^{m}}M = ∑_{j} a_{j}[L(j_{1})⊛⋯⊛L(j_{n})], summing over all j ∈ I^{n} with j_{n−m+1} = ⋯ = j_{n} = i.

Now for i ∈ I and M ∈ Rep_{I} H_{n}, define

ε_{i}(M) = max{m ≥ 0 | Δ_{i^{m}}M ≠ 0}.  (5.4)

Note Lemma 5.1 shows that ε_{i}(M) can be worked out just from knowledge of the character ch M.

Lemma 5.2. Let M ∈ Rep_{I} H_{n} be irreducible, i ∈ I, ε = ε_{i}(M). If N ⊛ L(i^{m}) is an irreducible submodule of Δ_{i^{m}}M for some 0 ≤ m ≤ ε, then ε_{i}(N) = ε − m. Proof. The definitions imply immediately that ε_{i}(N) ≤ ε − m. For the reverse inequality, (5.3) and the irreducibility of M gives that M is a quotient of ind_{n−m,m}^{n}N ⊛ L(i^{m}). So applying the exact functor Δ_{i^{ε}}, Δ_{i^{ε}}M ≠ 0 is a quotient of Δ_{i^{ε}}(ind_{n−m,m}^{n}N ⊛ L(i^{m})). In particular, Δ_{i^{ε}}(ind_{n−m,m}^{n}N ⊛ L(i^{m})) ≠ 0. Now one gets that ε_{i}(N) ≥ ε − m applying the Shuffle Lemma (Lemma 4.11) and Lemma 5.1.

Lemma 5.3. Let m ≥ 0, i ∈ I and N be an irreducible module in Rep_{I} H_{n} with ε_{i}(N) = 0. Set M := ind_{n,m}^{n+m}N ⊛ L(i^{m}). Then:

(i) Δ_{i^{m}}M ≅ N ⊛ L(i^{m}); (ii) cosoc M is irreducible with ε_{i}(cosoc M) = m; (iii) all other composition factors L of M have ε_{i}(L) < m.

Proof. (i) Clearly a copy of N ⊛ L(i^{m}) appears in Δ_{i^{m}}M. But by the Shuffle Lemma and Lemma 5.1, dim Δ_{i^{m}}M = dim N ⊛ L(i^{m}), hence Δ_{i^{m}}M ≅ N ⊛ L(i^{m}).

(ii) By (5.3), a copy of N ⊛ L(i^{m}) appears in Δ_{i^{m}}Q for any non-zero quotient Q of M, in particular for any constituent of cosoc M. But by (i), N ⊛ L(i^{m}) only appears once in Δ_{i^{m}}M, hence cosoc M must be irreducible.

(iii) We have shown that Δ_{i^{m}}M = Δ_{i^{m}}(cosoc M). Hence, Δ_{i^{m}}L = 0 for any other composition factor of M by exactness of Δ_{i^{m}}.

Lemma 5.4. Let M ∈ Rep_{I} H_{n} be irreducible, i ∈ I, ε = ε_{i}(M). Then, Δ_{i^{ε}}M is isomorphic to N ⊛ L(i^{ε}) for some irreducible H_{n−ε}-module N with ε_{i}(N) = 0. Proof. Pick an irreducible submodule of Δ_{i^{ε}}M. In view of Theorem 4.16(i), it must be of the form N ⊛ L(i^{ε}) for some irreducible H_{n−ε}-module N. Moreover, ε_{i}(N) = 0 by Lemma 5.2. By (5.3) and the irreducibility of M, M is a quotient of ind_{n−ε,ε}^{n}N ⊛ L(i^{ε}). Hence, Δ_{i^{ε}}M is a quotient of Δ_{i^{ε}}ind_{n−ε,ε}^{n}N ⊛ L(i^{ε}). But this is isomorphic to N ⊛ L(i^{ε}) by Lemma 5.3(i). This shows that Δ_{i^{ε}}M ≅ N ⊛ L(i^{ε}).

Lemma 5.5. Let m ≥ 0, i ∈ I and N be an irreducible module in Rep_{I} H_{n}. Set M := ind_{n,m}^{n+m}(N ⊛ L(i^{m})). Then, cosoc M is irreducible with ε_{i}(cosoc M) = ε_{i}(N) + m, and all other composition factors L of M have ε_{i}(L) < ε_{i}(N) + m. Proof. Let ε = ε_{i}(N). By Lemma 5.4, we have that Δ_{i^{ε}}N = K ⊛ L(i^{ε}) for an irreducible K ∈ Rep_{I} H_{n−ε} with ε_{i}(K) = 0. By (5.3) and the irreducibility of N, N is a quotient of ind_{n−ε,ε}^{n}K ⊛ L(i^{ε}). So the transitivity of induction implies that ind_{n,m}^{n+m}N ⊛ L(i^{m}) is a quotient of ind_{n−ε,ε+m}^{n+m}K ⊛ L(i^{ε+m}). Now everything follows from Lemma 5.3.

Theorem 5.6. Let M be an irreducible module in Rep_{I} H_{n} and i ∈ I. Then, for any 0 ≤ m ≤ ε_{i}(M), soc Δ_{i^{m}}M is an irreducible H_{n−m,m}-module of the same type as M, and is isomorphic to L ⊛ L(i^{m}) for some irreducible H_{n−m}-module L with ε_{i}(L) = ε_{i}(M) − m.

Proof. Let ε = ε_{i}(M). Suppose that K ⊛ L(i^{m}) is a constituent of soc Δ_{i^{m}}M. By Lemma 5.2, we have ε_{i}(K) = ε − m. So every such K contributes at least one composition factor to Δ_{i^{ε}}M. But Δ_{i^{ε}}M is irreducible by Lemma 5.4. Hence soc Δ_{i^{m}}M must be irreducible, too. Now certainly soc Δ_{i^{m}}M ≅ L ⊛ L(i^{m}) for some irreducible H_{n−m}-module L, by Theorem 4.16(i). It remains to show that L ⊛ L(i^{m}) has the same type as M. Note by Lemma 5.5, ind_{n−m,m}^{n}L ⊛ L(i^{m}) has irreducible cosocle, necessarily isomorphic to M by Frobenius reciprocity. So applying (5.3) we have that

End_{H_{n−m,m}}(L ⊛ L(i^{m})) ≃ Hom_{H_{n−m,m}}(L ⊛ L(i^{m}), Δ_{i^{m}}M)
≃ Hom_{H_{n}}(ind_{n−m,m}^{n}L ⊛ L(i^{m}), M) ≃ End_{H_{n}}(M),

which implies the statement concerning types.

The theorem has the following consequence:

Corollary 5.7. For irreducible M ∈ Rep_{I} H_{n}, the socle of res_{n−1,1}^{n}M is multiplicity-free.

We can also apply the theorem to study res_{n−1}^{n}M, meaning the restriction of M to the subalgebra H_{n−1} ⊂ H_{n}, see (2.27).

Corollary 5.8. For an irreducible M ∈ Rep_{I} H_{n} with ε_{i}(M) > 0,

soc res_{n−1}^{n−1,1} ∘ Δ_{i}(M) ≃ \{ \begin{matrix}L ⊕ ΠL & if M is of type Q or i ≠ 0, \\ L & if M is of type M and i = 0, \end{matrix}

for some irreducible H_{n−1}-module L of the same type as M if i ≠ 0 and of the opposite type to M if i = 0. Proof. Let δ := 1 if M is of type M and i = 0, δ := 2 otherwise. By Theorem 5.6, the socle of Δ_{i}M is isomorphic to L ⊛ L(i) for some irreducible H_{n−1}-module L, and

res_{n−1}^{n−1,1}L ⊛ L(i) ≅ L^{⊕δ},

indeed it is exactly as in the statement of the corollary. Now take any irreducible submodule K of res_{n−1}^{n−1,1} ∘ Δ_{i}(M). Consider the H_{n−1,1}-submodule H′_{1}K, where H′_{1} is the subalgebra generated by C_{n}, X_{n}^{±1}. All composition factors of H′_{1}K are isomorphic to K ⊛ L(i). In particular, the socle of H′_{1}K is isomorphic to K ⊛ L(i) which implies K ≅ L. We have now shown that soc res_{n−1}^{n−1,1}Δ_{i}(M) ≅ L^{⊕δ′} for some δ′ ≥ δ.

By Lemma 5.2, ε_{i}(L) = ε − 1 where ε = ε_{i}(M), and Δ_{i^{ε−1}}L is irreducible by Lemma 5.4. So at least δ′ copies of Δ_{i^{ε−1}}L appear in soc res_{n−ε,ε−1}^{n−ε,ε}Δ_{i^{ε}}(M). But Δ_{i^{ε}}M ≅ N ⊛ L(i^{ε}) for some irreducible N, so applying Theorem 4.16(iii) and the facts about type in Theorem 5.6,

soc res_{n−ε,ε−1}^{n−ε,ε}Δ_{i^{ε}}(M) ≅ soc res_{n−ε,ε−1}^{n−ε,ε}N ⊛ L(i^{ε})
≅ res_{n−ε,ε−1}^{n−ε,ε−1,1}N ⊛ L(i^{ε−1}) ⊛ L(i) ≅ (N ⊛ L(i^{ε−1}))^{⊕δ}.

Hence δ′ ≤ δ too.

§5-b. Operators ẽ_{i} and f̃_{i}. Let Irr_{I}H_{n} denote the set of isomorphism classes of irreducible modules in Rep_{I} H_{n}. We define

B(∞) = ⋃_{n≥0} Irr_{I}H_{n},  (5.5)

For each i ∈ I, we define the affine crystal operators

ẽ_{i} : B(∞) ∪ {0} → B(∞) ∪ {0},  (5.6)
f̃_{i} : B(∞) ∪ {0} → B(∞) ∪ {0}  (5.7)

as follows. First, we set ẽ_{i}(0) = f̃_{i}(0) = 0. Now let M be an irreducible module in Rep_{I} H_{n}. Then, f̃_{i}M is defined by

f̃_{i}M := cosoc ind_{n,1}^{n+1}M ⊛ L(i),  (5.8)

which is irreducible by Lemma 5.5. To define ẽ_{i}M, Theorem 5.6 shows that either Δ_{i}M = 0 or soc Δ_{i}M ≅ N ⊛ L(i) for an irreducible module N ∈ Rep_{I} H_{n−1}. In the former case, we define ẽ_{i}M = 0; in the latter case, we define ẽ_{i}M = N. Thus:

soc Δ_{i}M ≅ (ẽ_{i}M) ⊛ L(i).  (5.9)

Note right away from Lemma 5.2 that

ε_{i}(M) = max{m ≥ 0 | ẽ_{i}^{m}M ≠ 0},  (5.10)

while a special case of Lemma 5.5 shows that

ε_{i}(f̃_{i}M) = ε_{i}(M) + 1.  (5.11)

Lemma 5.9. Let M be an irreducible in Rep_{I} H_{n}, i ∈ I and m ≥ 0.

(i) soc Δ_{i^{m}}M ≅ (ẽ_{i}^{m}M) ⊛ L(i^{m}).

(ii) cosoc ind_{n,m}^{n+m}M ⊛ L(i^{m}) ≅ f̃_{i}^{m}M. Proof. (i) If m > ε_{i}(M), then both parts in the equality above are zero. Let m ≤ ε_{i}(M). By definition of ẽ_{i} and Lemma 5.2, (ẽ_{i}^{m}M)⊛L(i^{m}) is a submodule of Δ_{i^{m}}M. Now the result follows from Theorem 5.6.

(ii) By exactness of induction, f̃_{i}^{m}M is a quotient of ind_{n,m}^{n+m}M ⊛ L(i^{m}). Now the result follows from the simplicity of the cosocle, see Lemma 5.5.

Lemma 5.10. Let M ∈ Rep_{I} H_{n} and N ∈ Rep_{I} H_{n+1} be irreducible, and i ∈ I. Then, f̃_{i}M ≅ N if and only if ẽ_{i}N ≅ M. Proof. Suppose f̃_{i}M ≅ N. Then by (5.3), Hom_{H_{n,1}}(M ⊛ L(i), Δ_{i}N) ≠ 0, so M ⊛ L(i) appears in the socle of Δ_{i}N. This means that M ⊛ L(i) ≅ (ẽ_{i}N) ⊛ L(i), whence M ≅ ẽ_{i}N. The converse is similar.

Corollary 5.11. Let M, N be irreducible modules in Rep_{I} H_{n}. Then, f̃_{i}M ≅ f̃_{i}N if and only if M ≅ N. Similarly, providing ε_{i}(M), ε_{i}(N) > 0, ẽ_{i}M ≅ ẽ_{i}N if and only if M ≅ N.

We can also define cyclotomic analogues of the crystal operators. So now suppose that λ ∈ P_{+}, and let Irr H_{n}^{λ} denote the set of isomorphism classes of irreducible H_{n}^{λ}-modules. Define

B(λ) = ⋃_{n≥0} Irr H_{n}^{λ}.  (5.12)

The functors infl^{λ} and pr^{λ} induce maps

infl^{λ} : B(λ) ∪ {0} → B(∞) ∪ {0},  pr^{λ} : B(∞) ∪ {0} → B(λ) ∪ {0},  (5.13)

with pr^{λ} ∘ infl^{λ}(L) = L for each L ∈ B(λ). In other words, we can view B(λ) as a subset of B(∞) via the embedding infl^{λ}. Now by restricting ẽ_{i} and f̃_{i} to B(λ) ⊂ B(∞), we obtain the cyclotomic crystal operators, namely,

ẽ_{i}^{λ} = pr^{λ} ∘ ẽ_{i} ∘ infl^{λ} : B(λ) ∪ {0} → B(λ) ∪ {0},  (5.14)
f̃_{i}^{λ} = pr^{λ} ∘ f̃_{i} ∘ infl^{λ} : B(λ) ∪ {0} → B(λ) ∪ {0}  (5.15)

for each i ∈ I and λ ∈ P_{+}. Note that ẽ_{i} already maps B(λ) into B(λ) ∪ {0}, so we always have that ẽ_{i}M = ẽ_{i}^{λ}M for M ∈ B(λ). This is certainly not the case for f̃_{i}: for M ∈ B(λ), it will often be the case that f̃_{i}^{λ}(M) = 0 even though f̃_{i}(M) is never zero.

§5-c. Injectivity of ch. We can now prove an important theorem:

Theorem 5.12. The map ch : K(Rep_{I} H_{n}) → K(Rep_{I} A_{n}) is injective. Proof. We need to show that {ch L | [L] ∈ Irr_{I}H_{n}} is a linearly independent set in K(Rep_{I} A_{n}). Proceed by induction on n, the case n = 0 being trivial. Suppose n > 0 and

∑_{L∈Irr_{I}H_{n}} a_{L}ch L = 0

for some a_{L} ∈ ℤ. Choose any i ∈ I. We will show by downward induction on k = n, …, 1 that a_{L} = 0 for all L with ε_{i}(L) = k. Since every irreducible L has ε_{i}(L) > 0 for at least one i ∈ I, this is enough to complete the proof.

Consider first the case that k = n. Then, Δ_{i^{n}}L = 0 except if L ≅ L(i^{n}), by Theorem 4.16(i). Since ch Δ_{i^{n}}L can be worked out just from knowledge of ch L using Lemma 5.1, we deduce on applying Δ_{i^{n}} to the equation that the coefficient of ch L(i^{n}) is zero. Thus the induction starts. Now suppose 1 ≤ k < n and that we have shown a_{L} = 0 for all L with ε_{i}(L) > k. Apply Δ_{i^{k}} to the equation to deduce that

∑_{L with ε_{i}(L)=k} a_{L}ch Δ_{i^{k}}L = 0.

Now each such Δ_{i^{k}}L is irreducible, hence isomorphic to (ẽ_{i}^{k}L) ⊛ L(i^{k}), according to Lemmas 5.4 and 5.9(i). Moreover, for L ≅ L̸′, ẽ_{i}^{k}L ≅ ˜e̸_{i}^{k}L′ by Corollary 5.11. So now the induction hypothesis on n gives that all such coefficients a_{L} are zero, as required.

Corollary 5.13. If L is an irreducible module in Rep_{I} H_{n}, then L ≅ L^{τ}. Proof. Since τ(X_{i}) = X_{i}, τ leaves characters invariant. Hence it leaves irreducibles invariant since they are determined up to isomorphism by their character according to the theorem.

We can also show at this point that the type of an irreducible module L is determined by the type of its central character:

Lemma 5.14. Suppose L ∈ Rep_{I} H_{n} is irreducible with central character χ_{γ} where γ = ∑_{i∈I} γ_{i}α_{i} ∈ Γ_{n}. Then, L is of type Q is γ_{0} is odd, type M if γ_{0} is even. Proof. Proceed by induction on n, the case n = 0 being trivial. If n > 1, let L be an irreducible H_{n}-module with central character χ_{γ}. Choose i ∈ I so that ẽ_{i}L ≠ 0. By definition, ẽ_{i}L has central character γ − α_{i}. So by the induction hypothesis, ẽ_{i}L is of type Q if γ_{0} −δ_{i,0} is odd, type M otherwise. But by general theory §2-b and Lemma 4.7, (ẽ_{i}L)⊛L(i) is of the opposite type to ẽ_{i}L if i = 0, of the same type if i ≠ 0. Hence, (ẽ_{i}L) ⊛ L(i) is of type Q if γ_{0} is odd, type M otherwise. Finally, the proof is completed by Theorem 5.6, since this shows that L has the same type as soc Δ_{i}L = (ẽ_{i}L) ⊛ L(i).

§5-d. Crystal graphs. We can view the datum (B(∞), ẽ_{i}, f̃_{i}) as a combinatorial structure: the crystal graph. This is the directed graph with vertices the set B(∞) and an edge

[M] ⟶^{i} [N]

whenever [M], [N] ∈ B(∞) satisfy f̃_{i}M ≅ N, or equivalently, by Lemma 5.10, M ≅ ẽ_{i}N. Similarly, (B(λ), ẽ_{i}^{λ}, f̃_{i}^{λ}) can be viewed as a crystal graph.

Motivated by this, we introduce some notation to label the isomorphism classes of irreducible representations, or equivalently the vertices of the crystal graph. Write 1 for the (trivial) irreducible module of H_{0}. If L is an irreducible module in Rep_{I} H_{n}, one easily shows using Lemma 5.10 repeatedly that

L ≅ f̃_{i_{n}} … f̃_{i_{2}} f̃_{i_{1}}1

for at least one tuple i = (i_{1}, i_{2}, …, i_{n}) ∈ I^{n}. So if we define

L(i) = L(i_{1}, …, i_{n}) := f̃_{i_{n}} … f̃_{i_{2}} f̃_{i_{1}}1,

we obtain a labelling of all irreducibles in Rep_{I} H_{n} by tuples in I^{n}. For example, L(i, i, …, i) (n times) is precisely Kato’s module L(i^{n}). Similarly, any irreducible H_{n}^{λ}-module can be represented as

L^{λ}(i) = L^{λ}(i_{1}, …, i_{n}) := f̃_{i_{n}}^{λ}f̃_{i_{n−1}}^{λ} … f̃_{i_{1}}^{λ}1_{λ},

where 1_{λ} is the irreducible H_{0}^{λ}-module. Of course, L^{λ}(i) ≅ pr^{λ}L(i).

Thus, our labelling of the irreducible modules in Rep_{I} H_{n} is by paths in the crystal graph starting from 1, and similarly for Rep H_{n}^{λ}. Of course, the problem with this labelling is that a given irreducible L will in general be parametrized by several different tuples i ∈ I^{n}, corresponding to different paths from 1 to L. But basic properties of L(i) are easy to read off from the notation: for instance the central character of L(i) is χ_{i}, so by Lemma 5.14, L(i) is of type Q if an odd number of the i_{j} are zero, type M otherwise.

§5-e. Boring central characters. Given i = (i_{1}, …, i_{n}), let

ind(i) = ind(i_{1}, …, i_{n}) := ind_{1,...,1}^{n}L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}).

Note the character of ind(i) is ∑_{w∈S_{n}} L(i_{w^{−1}1}) ⊛ ⋯ ⊛ L(i_{w^{−1}n}). So every irreducible constituent of ind(i) belongs to the block Rep_{γ} H_{n}, where γ = wt(i).

Lemma 5.15. Let γ ∈ Γ_{n} and pick any i ∈ I^{n} with wt(i) = γ. Then:

(i) res_{1,...,1}^{n}L(i) has a submodule isomorphic to L(i_{1}) ⊛ ⋯ ⊛ L(i_{n}); (ii) ind(i) contains a copy of L(i) in its cosocle; (iii) every irreducible module in the block Rep_{γ} H_{n} appears at least once as a constituent

of ind(i). Proof. (i) Proceed by induction on n. For the induction step, let j = (i_{1}, …, i_{n−1}). By Frobenius reciprocity, there is a non-zero (hence necessarily injective) H_{n−1,1}-module homomorphism from L(j) ⊛ L(i_{n}) to res_{n−1,1}^{n}L(i). Hence by induction we get a copy of L(i_{1}) ⊛ ⋯ ⊛ L(i_{n−1}) ⊛ L(i_{n}) in res_{A_{n}}^{H_{n}}L(i).

(ii) Use (i) and Frobenius reciprocity.

(iii) We have just shown that L(i) appears in ind(i). But for any other j with the same weight as i, ind(j) has the same character as ind(i), hence they have the same set of composition factors thanks to Theorem 5.12. Hence, L(i) appears in ind(j).

We also need the following criterion for irreducibility, compare [G1, Lemma 5.9]:

Lemma 5.16. Let M, N be irreducibles in Rep_{I} H_{m}, Rep_{I} H_{n} respectively. Suppose (i) ind_{m,n}^{m+n}M ⊛ N ≅ ind_{n,m}^{n+m}N ⊛ M; (ii) M ⊛ N appears in res_{m,n}^{m+n}ind_{m,n}^{m+n}M ⊛ N with multiplicity one.

Then, ind_{m,n}^{m+n}M ⊛ N is irreducible. Proof. Suppose for a contradiction that K = ind_{m,n}^{m+n}M ⊛ N is reducible. Then we can find a proper irreducible submodule S, and set Q = K/S. By Frobenius reciprocity, M ⊛ N appears in res_{m,n}^{m+n}Q with non-zero multiplicity. Hence, it cannot appear in res_{m,n}^{m+n}S by assumption (ii). But assumption (i), Corollary 5.13 and Theorem 2.14 show that K ≅ K^{τ}. Hence, K also has a quotient isomorphic to S^{τ} ≅ S, and the Frobenius reciprocity argument implies that M ⊛ N appears in res_{m,n}^{m+n}S.

Lemma 5.17. Let i ∈ I^{m}, j ∈ I^{n} be tuples such that |i_{a} − j_{b}| > 1 for all 1 ≤ a ≤ m, 1 ≤ b ≤ n. Then, ind_{m,n}^{m+n}L(i) ⊛ L(j) ≅ ind_{n,m}^{m+n}L(j) ⊛ L(i) is irreducible. Proof. By Lemma 5.16 and the Shuffle Lemma, it suffices to show that

ind_{m,n}^{m+n}L(i) ⊛ L(j) ≅ ind_{n,m}^{m+n}L(j) ⊛ L(i).

By the Mackey Theorem, res_{m,n}^{m+n}ind_{n,m}^{m+n}L(j) ⊛ L(i) contains L(i) ⊛ L(j) as a summand with multiplicity one, all other constituents lying in different blocks. Hence by Frobenius reciprocity, there exists a non-zero homomorphism

f : ind_{m,n}^{m+n}L(i) ⊛ L(j) → ind_{n,m}^{m+n}L(j) ⊛ L(i).

Every homomorphic image of ind_{m,n}^{m+n}L(i)⊛L(j) contains an H_{m,n}-submodule isomorphic to L(i)⊛L(j). So, by Lemma 5.15(i), we see that the image of f contains an A_{m+n}-submodule V isomorphic to L(i_{1}) ⊛ ⋯ ⊛ L(i_{m}) ⊛ L(j_{1}) ⊛ ⋯ ⊛ L(j_{n}).

Now we claim that the image of f also contains an A_{m+n}-submodule isomorphic to L(j_{1}) ⊛ ⋯ ⊛ L(j_{n}) ⊛ L(i_{1}) ⊛ ⋯ ⊛ L(i_{m}). To see this, consider Φ̃_{m}V. Pick a common eigenvector v ∈ V for the operators X_{m}^{±1} and X_{m+1}^{±1}; then (X_{m} + X_{m}^{−1})v = q(i_{m})v and (X_{m+1} + X_{m+1}^{−1})v = q(j_{1})v. So according to (4.13), Φ̃_{m}^{2} acts on v by a scalar, and the assumption that |i_{m} −j_{1}| > 1 combined with Lemma 4.3 shows that this scalar is necessarily non-zero. Thus, Φ̃_{m}V ≠ 0, so by (4.15) it is an irreducible A_{m+n}-module, namely,

Φ̃_{m}V ≅ L(i_{1}) ⊛ ⋯ ⊛ L(i_{m−1}) ⊛ L(j_{1}) ⊛ L(i_{m}) ⊛ L(j_{2}) ⊛ ⋯ ⊛ L(j_{n}).

Next apply Φ̃_{m−1}, …, Φ̃_{1} to move L(j_{1}) to the first position, and continue in this way to complete the proof of the claim.

We have now shown that the image of f contains L(j_{1})⊛⋯⊛L(j_{n})⊛L(i_{1})⊛⋯⊛L(i_{m}). But by the Shuffle Lemma, all such composition factors of res_{1,...,1}^{m+n}ind_{n,m}^{m+n}L(j) ⊛ L(i) necessarily lie in the irreducible H_{n,m}-submodule 1 ⊗ L(j) ⊛ L(i) of the induced module. Since this generates all of ind_{n,m}^{m+n}L(j) ⊛ L(i) as an H_{m+n}-module, this shows that f is surjective. Hence f is an isomorphism by dimension, which completes the proof.

Theorem 5.18. Let i ∈ I^{m}, j ∈ I^{n} be tuples such that |i_{a} − j_{b}| ≠ 1 for all 1 ≤ a ≤ m, 1 ≤ b ≤ n. Then, ind_{m,n}^{m+n}L(i) ⊛ L(j) ≅ ind_{n,m}^{m+n}L(j) ⊛ L(i) is irreducible. Moreover, every other irreducible module lying in the same block as ind_{m,n}^{m+n}L(i) ⊛ L(j) is of the form ind_{m,n}^{m+n}L(i′) ⊛ L(j′) for permutations i′ of i and j′ of j. Proof. The second statement of the theorem is an easy consequence of the first and Lemma 5.15(iii). For the first statement, proceed by induction on m+n, the case m+n = 1 being trivial. For m + n > 1, we may assume by Lemma 5.17 that there exists k ∈ I that appears in both the tuples i and j. Note then that for every a = 1, …, m, either i_{a} = k or |i_{a} − k| > 1, and similarly for every b = 1, …, n, either j_{b} = k or |j_{b} − k| > 1. So by the induction hypothesis, we have that

L(i) ≅ ind_{n−r,r}^{n}L(i′) ⊛ L(k^{r}), L(j) ≅ ind_{m−s,s}^{m}L(j′) ⊛ L(k^{s})

for some r, s ≥ 1, where i′, j′ are tuples with no entries equal to k. By Theorem 2.14, Corollary 5.13 and Lemma 5.17,

ind_{r,m−s}^{r+m−s}L(k^{r}) ⊛ L(j′) ≅ ind_{m−s,r}^{m−s+r}L(j′) ⊛ L(k^{r}).

So using Theorem 4.16(i) and transitivity of induction,

ind_{m,n}^{m+n}L(i) ⊛ L(j) ≅ ind_{n−r,r,m−s,s}^{m+n}L(i′) ⊛ L(k^{r}) ⊛ L(j′) ⊛ L(k^{s})
≅ ind_{n−r,m−s,r+s}^{m+n}L(i′) ⊛ L(j′) ⊛ L(k^{r+s}).

Finally this is irreducible by the induction hypothesis and Lemma 5.17.

§5-f. Some character calculations. At this point we need to compute the characters of certain very special H_{n}-modules explicitly.

Lemma 5.19. Let i, j ∈ I with |i − j| = 1. Then, for all a, b ≥ 0 with a + b < −⟨h_{i}, α_{j}⟩, there is a non-split short exact sequence

0 ⟶ L(i^{a+1}ji^{b}) ⟶ ind_{a+b+1,1}^{a+b+2}L(i^{a}ji^{b}) ⊛ L(i) ⟶ L(i^{a}ji^{b+1}) ⟶ 0.

Moreover, for every a, b ≥ 0 with a + b ≤ −⟨h_{i}, α_{j}⟩,

ch L(i^{a}ji^{b}) = (a!)(b!)[L(i)^{⊛a} ⊛ L(j) ⊛ L(i)^{⊛b}].

Proof. We proceed by induction on n = 0, 1, …, −⟨h_{i}, α_{j}⟩ to show that

ch L(i^{n}j) = n![L(i)^{⊛n} ⊛ L(j)],

this being immediate in case n = 0. For n > 0, let M := ind_{n,1}^{n+1}L(i^{n−1}j) ⊛ L(i). We know by the inductive hypothesis and the Shuffle Lemma that

ch M = n![L(i)^{⊛n} ⊛ L(j)] + (n − 1)![L(i)^{⊛(n−1)} ⊛ L(j) ⊛ L(i)].

Now consider the H_{n,1}-submodule

N := (X_{n+1} + X_{n+1}^{−1} − q(i))M ≅ L(i^{n}) ⊛ L(j)

of M. The key point is that N is stable under the action of T_{n}, hence all of H_{n+1}. Although this is in principle an elementary calculation, it turns out to be extremely lengthy. It was carried out by hand for n ≤ 2 but for the cases n = 3, 4 (when ℓ necessarily equals 1), we had to resort to a computer calculation using the Gap computer algebra package. This proves the existence of an irreducible H_{n+1}-module N with character n![L(i)^{⊛n} ⊛ L(j)]. This must be L(i^{n}j), by Lemma 5.15(i), completing the proof of the induction step.

Now we explain how to deduce the characters of the remaining irreducibles in the block. In the argument just given, the quotient module M/N has character (n − 1)![L(i)^{⊛(n−1)} ⊛ L(j) ⊛ L(i)], so must be L(i^{n−1}ji). Twisting with the automorphism σ proves that there exist irreducibles with characters n![L(j) ⊛ L(i)^{⊛n}] and (n − 1)![L(i) ⊛ L(j) ⊛ L(i)^{⊛(n−1)}], which must be L(ji^{n}) and L(iji^{n−1}) respectively by Lemma 5.15(i) once more. This covers everything unless n = 4, when we necessarily have that i = 0, j = 1 and ℓ = 1. In this case, we have shown already that there exist four irreducibles with characters

ch L(00001) = 24[L(0)^{⊛4} ⊛ L(1)], ch L(00010) = 6[L(0)^{⊛3} ⊛ L(1) ⊛ L(0)],
ch L(10000) = 24[L(1) ⊛ L(0)^{⊛4}], ch L(01000) = 6[L(0) ⊛ L(1) ⊛ L(0)^{⊛3}].

So by Lemma 5.15, there must be exactly one more irreducible module in the block, namely L(00100), since none of the above involve the character [L(0)^{⊛2}⊛L(1)⊛L(0)^{⊛2}]. Considering the character of ind_{n,1}^{n+1}L(0010) ⊛ L(0) shows that ch L(00100) is either 4[L(0)^{⊛2} ⊛ L(1) ⊛ L(0)^{⊛2}] or 4[L(0)^{⊛2} ⊛ L(1) ⊛ L(0)^{⊛2}] + 6[L(0)^{⊛3} ⊛ L(1) ⊛ L(0)]. But the latter is not σ-invariant so cannot occur as there would then be too many irreducibles.

Now that the characters are known, it is finally a routine matter using the Shuffle Lemma and Lemma 5.5 to prove the existence of the required non-split sequence.

Lemma 5.20. Let i, j ∈ I with |i − j| = 1 and set n = 1 − ⟨h_{i}, α_{j}⟩. Then

L(i^{n}j) ≅ L(i^{n−1}ji).

Moreover, for every a, b ≥ 0 with a + b = −⟨h_{i}, α_{j}⟩,

L(i^{a}ji^{b+1}) ≅ ind_{n,1}^{n+1}L(i^{a}ji^{b}) ⊛ L(i) ≅ ind_{1,n}^{n+1}L(i) ⊛ L(i^{a}ji^{b})

with character a!(b + 1)![L(i)^{⊛a} ⊛ L(j) ⊛ L(i)^{⊛(b+1)}] + (a + 1)!b![L(i)^{⊛(a+1)} ⊛ L(j) ⊛ L(i)^{⊛b}]. Proof. Let M = ind_{n,1}^{n+1}L(i^{n−1}j) ⊛ L(i). We first claim that M is irreducible. To prove this, arguing in the same way as the proof of Lemma 5.19, it suffices to show that the H_{n,1}-submodule

(X_{n+1} + X_{n+1}^{−1} − q(i))M ≅ L(i^{n}) ⊛ L(j)

of M is not invariant under T_{n}. Again this was checked by an explicit computer calculation. Hence, there is an irreducible H_{n}-module M with character

n![L(i)^{⊛n} ⊛ L(j)] + (n − 1)![L(i)^{⊛(n−1)} ⊛ L(j) ⊛ L(i)].

Hence ẽ_{i}M ≅ L(i^{n−1}j) and ẽ_{j}M ≅ L(i^{n}) by Theorem 5.6. So we deduce that M ≅ L(i^{n−1}ji) ≅ L(i^{n}j) thanks to Lemma 5.10.

Now consider the remaining irreducibles in the block. There are at most (n−1) remaining, namely L(i^{a}ji^{b+1}) for a ≥ 0, b ≥ 1 with a+b = −⟨h_{i}, α_{j}⟩. Considering the known characters of ind_{n,1}^{n+1}L(i^{a}ji^{b}) ⊛ L(i) and arguing in a similar way to the second paragraph of the proof of the preceeding lemma, the remainder of the lemma follows without further calculation.

§5-g. Higher crystal operators. In this subsection we will introduce certain generalizations of the crystal operators f̃_{j}, following the ideas of [G1, §10]. The results of this subsection are only needed in §6-e below. To simplify notation, we will write simply ind in place of ind_{μ}^{n} throughout the subsection.

Lemma 5.21. Let i, j ∈ I with i ≠ j. For any a, b ≥ 0 with a + b = −⟨h_{i}, α_{j}⟩,

ind L(i^{a}ji^{b}) ⊛ L(i^{m}) ≅ ind L(i^{m}) ⊛ L(i^{a}ji^{b})

is irreducible.

Proof. We first claim that

ind L(i^{a}ji^{b}) ⊛ L(i^{m}) ≅ ind L(i^{m}) ⊛ L(i^{a}ji^{b}).

This is immediate from Lemma 5.17 in case |i − j| > 1. If |i − j| = 1, then transitivity of induction and Lemma 5.20 give that

ind L(i^{a}ji^{b}) ⊛ L(i^{m}) ≅ ind L(i^{a}ji^{b}) ⊛ L(i) ⊛ L(i^{m−1})
≅ ind L(i) ⊛ L(i^{a}ji^{b}) ⊛ L(i^{m−1}),

and now repeating this argument (m − 1) more times gives the claim.

Hence, by Corollary 5.13 and Theorem 2.14, K := ind L(i^{a}ji^{b}) ⊛ L(i^{m}) is self-dual. Now suppose for a contradiction that K is reducible. Then we can pick a proper irreducible submodule S of K, and set Q := K/S. Applying Lemmas 5.19 and 4.11,

ch K = ∑_{k=0}^{m} \binom{m}{k}(a + k)!(b + m − k)![L(i)^{⊛(a+k)} ⊛ L(j) ⊛ L(i)^{⊛(b+m−k)}].

By Frobenius reciprocity, Q contains an H_{a+b+1,m}-submodule isomorphic to L(i^{a}ji^{b})⊛L(i^{m}). So by Lemma 5.15(i), the irreducible A_{a+b+m+1}-module L(i)^{⊛a} ⊛ L(j) ⊛ L(i)^{⊛(b+m)} appears in Q with non-zero multiplicity, hence in fact by Theorem 4.16(i) it must appear with multiplicity a!(b + m)! (viewing Q as a module over H_{a,1,b+m}). It follows that L(i)^{⊛a} ⊛ L(j) ⊛ L(i)^{⊛(b+m)} is not a composition factor of S. But this is a contradiction, since as K is self-dual, S ≅ S^{τ} is a quotient module of K hence must contain L(i)^{⊛a} ⊛ L(j) ⊛ L(i)^{⊛(b+m)} by the Frobenius reciprocity argument again.

Lemma 5.22. Let i, j ∈ I with i ≠ j. For any a ≥ 1 and b ≥ 0 with a + b = −⟨h_{i}, α_{j}⟩, any irreducible module M in Rep_{I} H_{n} and any m ≥ 0,

cosoc ind M ⊛ L(i^{m}) ⊛ L(i^{a}ji^{b})

is irreducible. Proof. By the argument in the proof of Lemma 5.5, it suffices to prove this in the special case that ε_{i}(M) = 0. Let k = m + a + b + 1. Recall from the previous lemma that

N := ind L(i^{m}) ⊛ L(i^{a}ji^{b})

is an irreducible H_{k}-module. Moreover by Lemma 5.19, ch L(i^{a}ji^{b}) = (a!)(b!)[L(i)^{⊛a} ⊛ L(j) ⊛ L(j)^{⊛b}]. So since ε_{i}(M) = 0 and a > 0, the Mackey Theorem and a block argument shows that

res_{n,k}^{n+k}(ind M ⊛ L(i^{m}) ⊛ L(i^{a}ji^{b})) ≅ (M ⊛ N) ⊕ U

for some H_{n,k}-module U all of whose composition factors lie in different blocks to those of M ⊛ N. Now let H := cosoc ind M ⊛ L(i^{m}) ⊛ L(i^{a}ji^{b}). It follows from above that res_{n,k}^{n+k}H ≅ (M ⊛ N) ⊕ Ũ where Ũ is some quotient module of U. Then:

Hom_{H_{n+k}}(H, H) ≅ Hom_{H_{n+k}}(ind M ⊛ L(i^{m}) ⊛ L(i^{a}ji^{b}), H)
≅ Hom_{H_{n,k}}(M ⊛ N, res_{n,k}^{n+k}H)
≅ Hom_{H_{n,k}}(M ⊛ N, M ⊛ N ⊕ Ũ) ≅ Hom_{H_{n,k}}(M ⊛ N, M ⊛ N).

Since H is completely reducible and M ⊛ N is irreducible, this implies that H is irreducible too, as required.

Now we can define the higher crystal operators. Let i, j ∈ I with i ≠ j, and a ≥ 1, b ≥ 0 with a + b = −⟨h_{i}, α_{j}⟩. Then the special case m = 0 of the theorem shows that

f̃_{i^{a}ji^{b}}M := cosoc ind M ⊛ L(i^{a}ji^{b})

is irreducible for every irreducible M in Rep_{I} H_{n}. Thus we have defined an operator

f̃_{i^{a}ji^{b}} : B(∞) → B(∞).  (5.16)

Lemma 5.23. Take i, j ∈ I with i ≠ j and set k = −⟨h_{i}, α_{j}⟩. Let M be an irreducible module in Rep_{I} H_{n}.

(i) There exists a unique integer a with 0 ≤ a ≤ k such that for every m ≥ 0 we have

ε_{i}(f̃_{i}^{m}f̃_{j}M) = m + ε_{i}(M) − a.

(ii) Assume m ≥ k. Then a copy of f̃_{i}^{m}f̃_{j}M appears in the cosocle of

ind f̃_{i}^{m−k}M ⊛ L(i^{a}ji^{k−a}).

In particular, if a ≥ 1, then f̃_{i}^{m}f̃_{j}M ≅ f̃_{i^{a}ji^{k−a}} f̃_{i}^{m−k}M.

Proof. Let ε = ε_{i}(M) and write M = f̃_{i}^{ε}N for irreducible N ∈ Rep_{I} H_{n−ε} with ε_{i}(N) = 0. It suffices to prove (i) for any fixed choice of m, the conclusion for all other m ≥ 0 then following immediately by (5.11). So take m ≥ k. Note that f̃_{i}^{m}f̃_{j}M = f̃_{i}^{m}f̃_{j}f̃_{i}^{ε}N is a quotient of

ind N ⊛ L(i^{ε}) ⊛ L(j) ⊛ L(i)^{⊛k} ⊛ L(i^{m−k}),

which by Lemma 5.19 has a filtration with factors isomorphic to

F_{a} := ind N ⊛ L(i^{ε}) ⊛ L(i^{a}ji^{k−a}) ⊛ L(i^{m−k}), 0 ≤ a ≤ k.

So f̃_{i}^{m}f̃_{j}M is a quotient of some such factor, and to prove (i) it remains to show that ε_{i}(L) = ε + m − a for any irreducible quotient L of F_{a}. The inequality ε_{i}(L) ≤ ε + m − a is clear from the Shuffle Lemma. On the other hand, by transitivity of induction and Lemma 5.21, F_{a} ≅ ind N ⊛ (ind L(i^{a}ji^{k−a}) ⊛ L(i^{ε+m−k})). So by Frobenius Reciprocity, the irreducible module N ⊛ (ind L(i^{a}ji^{k−a}) ⊛ L(i^{ε+m−k})) is contained in res_{n−ε,m+1+ε}L. Hence ε_{i}(L) ≥ ε + m − a.

For (ii), by Lemma 5.21, we also have F_{a} ≅ ind N ⊛ L(i^{m−k+ε}) ⊛ L(i^{a}ji^{k−a}), and by the Shuffle Lemma, the only irreducible factors K of F_{a} with ε_{i}(K) = ε + m − a come from its quotient

ind f̃_{i}^{m−k+ε}N ⊛ L(i^{a}ji^{k−a}) ≅ indf̃_{i}^{m−k}M ⊛ L(i^{a}ji^{k−a}).

Finally, in case a ≥ 1, the cosocle of the last module is precisely f̃_{i^{a}ji^{k−a}} f̃_{i}^{m−k}M.

§5-h. Modifications in the degenerate case. Other than replacing X_{i} + X_{i}^{−1} by x_{i}^{2} everywhere, everything in this section goes through in the degenerate case in exactly the same way. Note the computer calculations in §5-f were checked separately in the degenerate case.

6. Induction and restriction

§6-a. i-induction and i-restriction. Fix λ ∈ P_{+} throughout the subsection. Recall the definition of the functors Δ_{i} for i ∈ I, see §5-a. Let us denote the composite functor res_{n−1}^{n−1,1} ∘ Δ_{i} instead by

res_{i} : Rep_{I} H_{n} → Rep_{I} H_{n−1},  (6.1)

for any n. Note if M is an H_{n}^{λ}-module, then res_{i}M is automatically an H_{n−1}^{λ}-module. So the restriction of the functor res_{i} gives a functor which we also denote

res_{i} : Rep H_{n}^{λ} → Rep H_{n−1}^{λ}.  (6.2)

We now focus on this cyclotomic case.

There is an alternative definition of res_{i}: if M is a module in Rep_{γ} H_{n}^{λ} for some fixed γ = ∑_{j∈I} γ_{j}α_{j} ∈ Γ_{n} then

res_{i}M = \{ \begin{matrix}(res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M)[γ − α_{i}] & if γ_{i} > 0, \\ 0 & if γ_{i} = 0. \end{matrix}  (6.3)

This description makes it clear how to define an analogous (additive) functor

ind_{i} : Rep H_{n}^{λ} → Rep H_{n+1}^{λ}.  (6.4)

Using (4.22) and additivity, it suffices to define this on an object M belonging to Rep_{γ} H_{n}^{λ} for fixed γ = ∑_{j∈I} γ_{j}α_{j} ∈ Γ_{n}. Then, we set

ind_{i}M = (ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M)[γ + α_{i}].  (6.5)

By the definitions (6.3) and (6.5) and Lemma 4.9, we have that

ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M = ⨁_{i∈I} ind_{i}M,  res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M = ⨁_{i∈I} res_{i}M.  (6.6)

To complete the definition of the functor ind_{i}, it is defined on a morphism f simply by

restriction of the corresponding morphism ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}f. We stress that the functor ind_{i} depends fundamentally on the fixed choice of λ, unlike res_{i} which is just the restriction of its affine counterpart.

Lemma 6.1. For λ ∈ P_{+} and each i ∈ I,

(i) ind_{i} and res_{i} are both left and right adjoint to each other, hence they are exact and send projectives to projectives;

(ii) ind_{i} and res_{i} commute with duality, i.e. there are natural isomorphisms

ind_{i}(M^{τ}) ≃ (ind_{i}M)^{τ} , res_{i}(M^{τ}) ≃ (res_{i}M)^{τ}

for each finite dimensional H_{n}^{λ}-module M.

Proof. We know that ind_{i}M and res_{i}M are summands of ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M and res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M

respectively. Moreover, τ-duality leaves central characters invariant because τ(X_{j}) = X_{j} for each j. Now everything follows easily applying Corollary 3.15.

In order to refine the definitions of ind_{i} and res_{i} in the next subsection, we need to give an alternative definition, due to Grojnowski [G1, §8] in the untwisted case. Recall the definition of the left H_{1}-modules R_{m}(i) for i ∈ I, m ≥ 0 from (4.23). The limits in the next lemma are taken with respect to the systems induced by the maps (4.24).

Lemma 6.2. For every finite dimensional H_{n}^{λ}-module M and i ∈ I, there are natural isomorphisms

ind_{i}M ≃ \underleftarrow{lim} pr^{λ}ind_{n,1}^{n+1}M ⊠ R_{m}(i),
res_{i}M ≃ \underrightarrow{lim} pr^{λ}Hom_{H′_{1}}(R_{m}(i), M)

(in the second case, H′_{1} denotes the subalgebra of H_{n−1,1} generated by C_{n}, X_{n}^{±1} and the H_{n−1}-module structure is defined by (hf)(r) = h(f(r)) for f ∈ Hom_{H′_{1}}(R_{m}(i), M) and r ∈ R_{m}(i)). Proof. For res_{i}, it suffices to consider the effect on M ∈ Rep_{γ} H_{n}^{λ} for γ = ∑_{j∈I} γ_{j}α_{j} ∈ Γ_{n} with γ_{i} > 0, both sides of what we are trying to prove clearly being zero if γ_{i} = 0. Then, for all sufficiently large m, Lemma 4.18 (in the special case n = 1) implies that

Hom_{H′_{1}}(R_{m}(i), M) ≃ (res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M)[γ − α_{i}].

Hence,

\underrightarrow{lim} pr^{λ}Hom_{H′_{1}}(R_{m}(i), M) ≃ (res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M)[γ − α_{i}] = res_{i}M.

This proves the lemma for res_{i}.

To deduce the statement about induction, it now suffices by uniqueness of adjoint functors to show that \underleftarrow{lim} pr^{λ}ind_{n,1}^{n+1}? ⊠ R_{m}(i) is left adjoint to \underrightarrow{lim} pr^{λ}Hom_{H′_{1}}(R_{m}(i),?). Let N ∈ Rep H_{n−1}^{λ} and M ∈ Rep H_{n}^{λ}. First observe as explained in the previous paragraph that the direct system

pr^{λ}Hom_{H′_{1}}(R_{1}(i), M) ↪ pr^{λ}Hom_{H′_{1}}(R_{2}(i), M) ↪ …

stabilizes after finitely many terms. We claim that the inverse system

pr^{λ}ind_{n,1}^{n+1}N ⊠ R_{1}(i) ↞ pr^{λ}ind_{n,1}^{n+1}N ⊠ R_{2}(i) ↞ …

also stabilizes after finitely many terms. To see this, it suffices to show that the dimension of pr^{λ}ind_{n,1}^{n+1}N ⊠R_{m}(i) is bounded above independently of m. Well, each R_{m}(i) is generated as an H′_{1}-module by a subspace W isomorphic (as a vector space) to the cosocle R_{1}(i) of R_{m}(i). Then ind_{n,1}^{n+1}N ⊠R_{m}(i) is generated as an H_{n+1}-module by the subspace W′ = 1⊗(N ⊗W), also of dimension independent of m. Finally, pr^{λ}ind_{n,1}^{n+1}N ⊠R_{m}(i) is a quotient of the vector space H_{n+1}^{λ} ⊗_{F} W′, whose dimension is independent of m.

Now we can complete the proof of adjointness. Using the fact from the previous paragraph that the direct and inverse systems stabilize after finitely many terms, we have natural isomorphisms

Hom_{H_{n}^{λ}}(\underleftarrow{lim} pr^{λ}ind_{n−1,1}^{n}N ⊠ R_{m}(i), M) ≃ \underrightarrow{lim} Hom_{H_{n}^{λ}}(pr^{λ}ind_{n−1,1}^{n}N ⊠ R_{m}(i), M)
≃ \underrightarrow{lim} Hom_{H_{n}}(ind_{n−1,1}^{n}N ⊠ R_{m}(i), M)
≃ \underrightarrow{lim} Hom_{H_{n−1,1}}(N ⊠ R_{m}(i), res_{n−1,1}^{n}M)
≃ \underrightarrow{lim} Hom_{H_{n−1}}(N, Hom_{H′_{1}}(R_{m}(i), M))
≃ \underrightarrow{lim} Hom_{H_{n−1}^{λ}}(N, pr^{λ}Hom_{H′_{1}}(R_{m}(i), M))
≃ Hom_{H_{n−1}^{λ}}(N, \underrightarrow{lim} pr^{λ}Hom_{H′_{1}}(R_{m}(i), M)).

This completes the argument.

§6-b. Operators e_{i} and f_{i}. Continue with λ ∈ P_{+} being fixed. We wish to refine the definitions of the functors res_{i} and ind_{i} to give operators, denoted e_{i} and f_{i} respectively, from irreducible H_{n}^{λ}-modules to isomorphism classes of H_{n−1}^{λ}- (resp. H_{n+1}^{λ}-) modules.

Actually, e_{i} is simply the restriction to Irr H_{n}^{λ} of an operator also denoted e_{i} on the irreducible modules in Rep_{I} H_{n}. We define this first; recall the definition of the module L_{m}(i) from §4-h. Let M be an irreducible in Rep_{I} H_{n}. Let H′_{1} denote the subalgebra of H_{n} generated by C_{n}, X_{n}^{±1}. For each m ≥ 1, we define an H_{n−1}-module

\overline{Hom}_{H′_{1}}(L_{m}(i), M)  (6.7)

as follows. If M is of type M or i ≠ 0, this is simply the space Hom_{H′_{1}}(L_{m}(i), M) viewed as an H_{n−1}-module in the same way as in Lemma 6.2. But if M is of type Q and i = 0, we can pick an odd involution θ_{M} : M → M and also have the odd involutions θ_{m} : L_{m}(i) → L_{m}(i) from (4.26). Let

θ_{M} ⊗ θ_{m} : Hom_{H′_{1}}(L_{m}(i), M) → Hom_{H′_{1}}(L_{m}(i), M)

denote the map defined by ((θ_{M} ⊗ θ_{m})f)(v) = (−1)f̄θ_{M}(f(θ_{m}v)). One checks that (θ_{M} ⊗ θ_{m})^{2} = 1, hence the ±1-eigenspaces of θ_{M} ⊗ θ_{m} split Hom_{H′_{1}}(L_{m}(i), M) into a direct sum of two isomorphic H_{n−1}-modules (because there is an obvious odd automorphism swapping the two eigenspaces). Now in this case, we define \overline{Hom}_{H′_{1}}(L_{m}(i), M) to be the 1-eigenspace (say).

In either case, we have a direct system

\overline{Hom}_{H′_{1}}(L_{1}(i), M) ↪ \overline{Hom}_{H′_{1}}(L_{2}(i), M) ↪ …

induced by the inverse system (4.25). Now define

e_{i}M = \underrightarrow{lim} \overline{Hom}_{H′_{1}}(L_{m}(i), M),  (6.8)

giving us the affine version of the operator e_{i}. Note if M is an H_{n}^{λ}-module then each \overline{Hom}_{H′_{1}}(L_{m}(i), M) is an H_{n−1}^{λ}-module, so

e_{i}M = \underrightarrow{lim} pr^{λ}\overline{Hom}_{H′_{1}}(L_{m}(i), M).  (6.9)

We take (6.9) as our definition of the operator e_{i} in the cyclotomic case. Comparing (6.9) with Lemma 6.2 and using (4.28), one sees at once that:

Lemma 6.3. Let i ∈ I and M be an irreducible module in Rep_{I} H_{n}, or an irreducible H_{n}^{λ}-module. Then,

res_{i}M ≃ \{ \begin{matrix}e_{i}M & if i = 0 and M is of type M, \\ e_{i}M ⊕ Πe_{i}M & otherwise. \end{matrix}

Now we turn to the definition of f_{i}M which, just like ind_{i}M, only makes sense in the cyclotomic case. So, let M an irreducible H_{n}^{λ}-module. We need to extend the definition of the operation ⊛ to give meaning to the notation M ⊛ L_{m}(i), for each m ≥ 1. If either M is of type M or i ≠ 0, then M ⊛ L_{m}(i) := M ⊠ L_{m}(i). But if M is of type Q and i = 0, pick an odd involution θ_{M} : M → M. Then, the ±\sqrt{−1}-eigenspaces of θ_{M} ⊗ θ_{m} acting on the left on M ⊠ L_{m}(i) split it into a direct sum of two isomorphic H_{n,1}-modules. Let M ⊛ L_{m}(i) denote the \sqrt{−1}-eigenspace (say) for each m.

We then have an inverse system M ⊛ L_{1}(i) ↞ M ⊛ L_{2}(i) ↞ … of H_{n,1}-modules induced by the maps from (4.25). Now we can define

f_{i}M = \underleftarrow{lim} pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i).  (6.10)

Comparing the definition with the proof of Lemma 6.2, one sees that the inverse limit stabilizes after finitely many terms, hence that f_{i}M really is a well-defined finite dimensional H_{n+1}^{λ}-module. Indeed:

Lemma 6.4. Let i ∈ I and M be an irreducible H_{n}^{λ}-module. Then,

ind_{i}M ≃ \{ \begin{matrix}f_{i}M & if i = 0 and M is of type M, \\ f_{i}M ⊕ Πf_{i}M & otherwise. \end{matrix}

Lemma 6.5. Let i ∈ I and M be an irreducible module in Rep_{I} H_{n}. Then, e_{i}M is non-zero if and only if ẽ_{i}M ≠ 0, in which case it is a self-dual indecomposable module with irreducible socle and cosocle isomorphic to ẽ_{i}M. Proof. To see that e_{i}M has irreducible socle ẽ_{i}M whenever it is non-zero, combine Lemma 6.3 with Corollary 5.8. The remaining facts follow since M is self-dual by Lemma 5.13, and res_{i} commutes with duality by Lemma 6.1(ii).

Before the next theorem, we recall again that the operator f_{i} depends critically on the fixed choice of λ. Theorem 6.6. Let λ ∈ P_{+} and i ∈ I. Then, for any irreducible H_{n}^{λ}-module M,

(i) e_{i}M is non-zero if and only if ẽ_{i}^{λ}M ≠ 0, in which case it is a self-dual indecomposable module with irreducible socle and cosocle isomorphic to ẽ_{i}^{λ}M;

(ii) f_{i}M is non-zero if and only if f̃_{i}^{λ}M ≠ 0, in which case it is a self-dual indecomposable

module with irreducible socle and cosocle isomorphic to f̃_{i}^{λ}M. Proof. (i) This is immediate from Lemma 6.5.

(ii) We deduce this from (i) by an adjointness argument. Let M be an irreducible H_{n}^{λ}- module, and N be an irreducible H_{n+1}^{λ}-module. Let δ_{M} equal 1 if i = 0 and M is of type M, 2 otherwise, and define δ_{N} similarly. Then, by Lemmas 6.1(i), 6.3 and 6.4,

dim Hom_{H_{n+1}^{λ}}(f_{i}M, N) = \frac{1}{δ_{M}} dim Hom_{H_{n+1}^{λ}}(ind_{i}M, N)

= \frac{1}{δ_{M}} dim Hom_{H_{n}^{λ}}(M, res_{i}N) = \frac{δ_{N}}{δ_{M}} dim Hom_{H_{n}^{λ}}(M, e_{i}N).

By (i), the latter is zero unless M = ẽ_{i}N, or equivalently N = f̃_{i}M by Lemma 5.10. Taking into account the superalgebra analogue of Schur’s lemma using Lemma 5.14, one deduces that cosoc f_{i}M ≅ f̃_{i}M. Finally, note f_{i}M is self-dual by Lemma 6.1(ii) so everything else follows.

Remark 6.7. Let us also point out, as follows easily from the definitions, that e_{i}M and f_{i}M admit odd involutions if either i ≠ 0 and M is of type Q, or i = 0 and M is of type M.

§6-c. Divided powers. Continue with λ ∈ P_{+} and fix i ∈ I. We can generalize the definitions of e_{i}, f_{i} to define operators denoted e_{i}^{(r)}, f_{i}^{(r)} on irreducible H_{n}^{λ}-modules, for each r ≥ 1. It will be the case that e_{i}^{(1)} = e_{i}, f_{i}^{(1)} = f_{i}. For the definitions, we make use of the covering modules L_{m}(i^{r}) from §4-h.

Let M be an irreducible module in Rep_{I} H_{n}. If r > n, we set e_{i}^{(r)}M = 0. Otherwise, let H′_{r} denote the subalgeba of H_{n} generated by X_{n−r+1}^{±1}, …, X_{n}^{±1}, C_{n−r+1}, …, C_{n}, T_{n−r+1}, …, T_{n−1}. We have a direct system

\overline{Hom}_{H′_{r}}(L_{1}(i^{r}), M) ↪ \overline{Hom}_{H′_{r}}(L_{2}(i^{r}), M) ↪ …

induced by the inverse system (4.25), where \overline{Hom} is interpreted in exactly the same way as in §6-b using the generalized maps θ_{m} from (4.26) in case i = 0. Now define

e_{i}^{(r)}M = \underrightarrow{lim} \overline{Hom}_{H′_{r}}(L_{m}(i^{r}), M).  (6.11)

As in §6-b, if M is an H_{n}^{λ}-module then e_{i}^{(r)}M is too, so that

e_{i}^{(r)}M = \underrightarrow{lim} pr^{λ}\overline{Hom}_{H′_{1}}(L_{m}(i^{r}), M)  (6.12)

in the cyclotomic case.

To define f_{i}^{(r)}, which as usual only makes sense in the cyclotomic case, let M be an irreducible H_{n}^{λ}-module. We have an inverse system M ⊛ L_{1}(i^{r}) ↞ M ⊛ L_{2}(i^{r}) ↞ … of H_{n,r}-modules induced by the maps from (4.25), again interpreting ⊛ as in §6-b. Now we can define

f_{i}^{(r)}M = \underleftarrow{lim} pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i^{r}).  (6.13)

Lemma 6.8. Let i ∈ I, r ≥ 1 and M be an irreducible H_{n}^{λ}-module. Then:

(res_{i})^{r}M ≃ \{ \begin{matrix}(e_{i}^{(r)}M ⊕ Πe_{i}^{(r)}M)^{⊕2^{r−1}(r!)} & if i ≠ 0, \\ e_{i}^{(r)}M^{⊕2^{(r−1)/2}(r!)} & if i = 0, r is odd, M is of type M, \\ (e_{i}^{(r)}M ⊕ Πe_{i}^{(r)}M)^{⊕2^{⌊(r−1)/2⌋}(r!)} & otherwise; \end{matrix}

(ind_{i})^{r}M ≃ \{ \begin{matrix}(f_{i}^{(r)}M ⊕ Πf_{i}^{(r)}M)^{⊕2^{r−1}(r!)} & if i ≠ 0, \\ f_{i}^{(r)}M^{⊕2^{(r−1)/2}(r!)} & if i = 0, r is odd, M is of type M, \\ (f_{i}^{(r)}M ⊕ Πf_{i}^{(r)}M)^{⊕2^{⌊(r−1)/2⌋}(r!)} & otherwise. \end{matrix}

Proof. Using Lemma 4.17 and the definitions, it suffices to show that

(res_{i})^{r}M ≃ \underrightarrow{lim} pr^{λ}Hom_{H′_{r}}(R_{m}(i^{r}), M),
(ind_{i})^{r}M ≃ \underleftarrow{lim} pr^{λ}ind_{n,r}^{n+r}M ⊠ R_{m}(i^{r}).

For (res_{i})^{r}, this follows from Lemma 4.18 in exactly the same way as in the proof of Lemma 6.2. Now (ind_{i})^{r} is left adjoint to (res_{i})^{r}, so the statement for induction follows from uniqueness of adjoint functors on checking that the functor \underleftarrow{lim} pr^{λ}ind_{n,r}^{n+r}? ⊠ R_{m}(i^{r}) is left adjoint to \underrightarrow{lim} pr^{λ}Hom_{H′_{r}}(R_{m}(i^{r}),?). The latter follows as in the proof of Lemma 6.2.

Since we have defined the operator e_{i}^{(r)} on irreducible modules we get induced operators also denoted e_{i}^{(r)} at the level of Grothendieck groups, namely,

e_{i}^{(r)} : K(Rep_{I} H_{n}) → K(Rep_{I} H_{n−r}), e_{i}^{(r)} : K(Rep H_{n}^{λ}) → K(Rep H_{n−r}^{λ}),

in the affine and cyclotomic cases respectively. Similarly f_{i}^{(r)} induces an operator

f_{i}^{(r)} : K(Rep H_{n}^{λ}) → K(Rep H_{n+r}^{λ})

on Grothendieck groups. We record:

Lemma 6.9. As operators on the Grothendieck group K(Rep H_{n}^{λ}) (or on K(Rep_{I} H_{n}) in the case of e_{i}), we have that e_{i}^{r} = (r!)e_{i}^{(r)} and f_{i}^{r} = (r!)f_{i}^{(r)}. Proof. Let us prove in the case i ≠ 0. The proof for i = 0 is the same idea, though the details are more delicate (one needs to use Lemma 5.14 too). By Lemmas 6.3 and 6.4, we have that

(res_{i})^{r} = 2^{r}e_{i}^{r}, (ind_{i})^{r} = 2^{r}f_{i}^{r}

as operators on the Grothendieck group. By Lemma 6.8, we have that

(res_{i})^{r} = 2^{r}(r!)e_{i}^{(r)}, (ind_{i})^{r} = 2^{r}(r!)f_{i}^{(r)}.

This is enough.

Let us finally note that we have only defined the operators e_{i}^{(r)} and f_{i}^{(r)} on irreducible modules. However, the definitions could be made more generally on pairs (M, θ_{M}), where M is an H_{n}^{λ}-module (or an integral H_{n}-module in the case of e_{i}) and θ_{M} : M → M is either the identity map or else an odd involution of M. In case θ_{M} = id_{M}, the definitions of e_{i}^{(r)}M and f_{i}^{(r)}M are exactly the same as the case M is irreducible of type M above. In case θ_{M} is an odd involution, the definitions of e_{i}^{(r)}M and f_{i}^{(r)}M are exactly the same as the case M is irreducible of type Q above, substiting the given map θ_{M} for the canonical odd involution of M in the situation above.

This remark applies especially to give us modules e_{i}^{(r)}P_{M}, f_{i}^{(r)}P_{M}, where P_{M} is the projective cover of an irreducible H_{n}^{λ}-module: in this case, if M is of type Q, the odd involution θ_{M} of M lifts to a unique odd involution also denoted θ_{M} of the projective cover. On doing this, we have that

[e_{i}^{(r)}P_{M}] = e_{i}^{(r)}[P_{M}],  [f_{i}^{(r)}P_{M}] = f_{i}^{(r)}[P_{M}]  (6.14)

where the equalities are written in K(Rep H_{n−r}^{λ}) and K(Rep H_{n+r}^{λ}) respectively. To prove this, one needs to observe that all composition factors of P_{M} are of the same type as M by Lemma 5.14.

Note Lemma 6.8 is also true if M is replaced by its projective cover P_{M}, the proof being the same as above. In particular, this shows that e_{i}^{(r)}P_{M} is a summand of (res_{i})^{r}P_{M}, and similarly for f_{i}. So Lemma 6.1(i) gives that e_{i}^{(r)}P_{M} and f_{i}^{(r)}P_{M} are also projective modules. Hence e_{i}^{(r)} and f_{i}^{(r)} induce operators with the same names on the Grothendieck groups of projective modules too:

e_{i}^{(r)} : K(Proj H_{n}^{λ}) → K(Proj H_{n−r}^{λ}), f_{i}^{(r)} : K(Proj H_{n}^{λ}) → K(Proj H_{n+r}^{λ}).

Moreover, by the same argument as in the proof of Lemma 6.9, we have:

Lemma 6.10. As operators on the Grothendieck group K(Proj H_{n}^{λ}), we have that

e_{i}^{r}[P_{M}] = (r!)[e_{i}^{(r)}P_{M}], f_{i}^{r}[P_{M}] = (r!)[f_{i}^{(r)}P_{M}],

for all irreducible H_{n}^{λ}-modules M.

§6-d. Alternative descriptions of ε_{i}. In this subsection we give three new interpretations of the functions ε_{i}, precisely as in [G1, Theorem 9.13].

Theorem 6.11. Let i ∈ I and M be an irreducible module in Rep_{I} H_{n}, Then (i) [e_{i}M] = ε_{i}(M)[ẽ_{i}M]+∑ c_{a}[N_{a}] where the N_{a} are irreducibles with ε_{i}(N_{a}) < ε_{i}(ẽ_{i}M); (ii) ε_{i}(M) is the maximal size of a Jordan block of X_{n} + X_{n}^{−1} (resp. X_{n} if i = 0) on M

with eigenvalue q(i) (resp. eigenvalue 1 if i = 0); (iii) End_{H_{n−1}}(e_{i}M) ≃ End_{H_{n−1}}(ẽ_{i}M)^{⊕ε_{i}(M)} as vector superspaces.

Proof. Let ε = ε_{i}(M) and N = ẽ_{i}^{ε}M.

(i) By Lemma 5.4 and Frobenius reciprocity, there is a short exact sequence

0 ⟶ R ⟶ ind_{n−ε,ε}^{n}N ⊛ L(i^{ε}) ⟶ M ⟶ 0.

Moreover, all composition factors L of R have ε_{i}(L) < ε by Lemma 5.3(iii). Applying the exact functor Δ_{i}, we obtain an exact sequence

0 ⟶ Δ_{i}R ⟶ Δ_{i}ind_{n−ε,ε}N ⊛ L(i^{ε}) ⟶ Δ_{i}M ⟶ 0.

By the Mackey Theorem, Δ_{i}ind_{n−ε,ε}^{n}N ⊛ L(i^{ε}) ≅ ind_{n−ε,ε−1,1}^{n−1,1}N ⊛ Δ_{i}L(i^{ε}). By considering characters [Δ_{i}L(i^{ε})] = ε[L(i^{ε−1}) ⊛ L(i)]. Hence,

[Δ_{i}ind_{n−ε,ε}^{n}N ⊛ L(i^{ε})] = ε[ind_{n−ε,ε−1,1}^{n−1,1}N ⊛ L(i^{ε−1}) ⊛ L(i)].  (6.15)

Using Lemma 5.3 again, the cosocle of ind_{n−ε,ε−1,1}^{n−1,1}N ⊛ L(i^{ε−1}) ⊛ L(i) is (ẽ_{i}M) ⊛ L(i), and all other composition factors of this module are of the form L ⊛ L(i) with ε_{i}(L) < ε − 1. Moreover, all composition factors of Δ_{i}R are of the form L ⊛ L(i) with ε_{i}(L) < ε − 1. So we have now shown that

[Δ_{i}M] = ε[ẽ_{i}M ⊛ L(i)] + ∑ c_{a}[N_{a} ⊛ L(i)]

for irreducibles N_{a} with ε_{i}(N_{a}) < ε_{i}(ẽ_{i}M). The conclusion follows on applying Lemma 6.3.

(ii) We give the argument for the case i ≠ 0, the case i = 0 being similar but using Lemma 4.15(ii) instead of Lemma 4.15(i). We know that Δ_{i^{ε}}M ≅ N ⊛ L(i^{ε}). So, applying the automorphism σ to Lemma 4.15(i), we deduce that the maximal size of a Jordan block of X_{n} + X_{n}^{−1} on Δ_{i^{ε}}M is ε. Hence the maximal size of a Jordan block of X_{n} + X_{n}^{−1} on Δ_{i}M is at least ε.

On the other hand, the argument given above in deriving (6.15) shows that the module Δ_{i}ind_{n−ε,ε}^{n}N ⊛ L(i^{ε}) has a filtration with ε factors, each of which is isomorphic to ind_{n−ε,ε−1,1}^{n−1,1}N ⊛L(i^{ε−1})⊛L(i). Since (X_{n}+X_{n}^{−1}−q(i)) annihilates ind_{n−ε,ε−1,1}^{n−1,1}N ⊛L(i^{ε−1})⊛ L(i), it follows that (X_{n} + X_{n}^{−1} − q(i))^{ε} annihilates Δ_{i}ind_{n−ε,ε}^{n}N ⊛ L(i^{ε}). So certainly (X_{n} + X_{n}^{−1} − q(i))^{ε} annihilates its quotient Δ_{i}M. So the maximal size of a Jordan block of X_{n} + X_{n}^{−1} on Δ_{i}M is at most ε.

(iii) Let z = (X_{n} + X_{n}^{−1} − q(i)) if i ≠ 0 and (X_{n} − 1) if i = 0. Consider the effect of left multiplication by z on the H_{n−1}-module R = res_{n−1}^{n−1,1} ∘ Δ_{i}(M). Note R is equal to either e_{i}M or e_{i}M ⊕ Πe_{i}M, by Lemma 6.3. In the latter case, (X_{n} + X_{n}^{−1}) (resp. X_{n}) acts as a scalar on soc R ≃ ẽ_{i}M ⊕Πẽ_{i}M, hence it leaves the two indecomposable summands invariant. This shows that in any case left multiplication by z (which centralizes the subalgebra H_{n−1} of H_{n}) induces an H_{n−1}-endomorphism θ : e_{i}M → e_{i}M. But by (ii), θ^{ε−1} ≠ 0 and θ^{ε} = 0. Hence, 1, θ, …, θ^{ε−1} give ε linearly independent even H_{n−1}-endomorphisms of e_{i}M. In view of Remark 6.7, we automatically get from these ε linearly independent odd endomorphisms in case ẽ_{i}M is of type Q, so we have now shown that

dim End_{H_{n−1}}(e_{i}M) ≥ ε dim End_{H_{n−1}}(ẽ_{i}M).

On the other hand, e_{i}M has irreducible cosocle ẽ_{i}M, and this appears in e_{i}M with multiplicity ε by (i), so the reverse inequality also holds.

Corollary 6.12. Let M, N be irreducible H_{n}-modules with M ≅ N̸. Then, for every i ∈ I, Hom_{H_{n−1}}(e_{i}M, e_{i}N) = 0. Proof. Suppose there is a non-zero homomorphism θ : e_{i}M → e_{i}N. Then, since e_{i}M has simple head ẽ_{i}M, we see that e_{i}N has ẽ_{i}M as a composition factor. Hence, by Theorem 6.11(i), ε_{i}(ẽ_{i}N) ≥ ε_{i}(ẽ_{i}M). Dualizing and applying the same argument gives the inequality the other way round, hence ε_{i}(ẽ_{i}N) = ε_{i}(ẽ_{i}M). But then, ẽ_{i}M is a composition factor of e_{i}N with ε_{i}(ẽ_{i}M) = ε_{i}(ẽ_{i}N), hence by Theorem 6.11(i) again, ẽ_{i}M ≅ ẽ_{i}N. But this contradicts Corollary 5.11.

To state the next corollary, we first need to introduce the ∗-operation on the crystal graph. This will play a fundamental role later on. Suppose M is an irreducible module in Rep_{I} H_{n} and 0 ≤ m ≤ n. Using Lemma 2.9 for the second equality in (6.18), define

ẽ_{i}^{∗}M = (ẽ_{i}(M^{σ}))^{σ},  (6.16)
f̃_{i}^{∗}M = (f̃_{i}(M^{σ}))^{σ} = cosoc ind_{1,n}^{n+1}L(i) ⊛ M,  (6.17)
ε_{i}^{∗}(M) = ε_{i}(M^{σ}) = max{m ≥ 0 | (ẽ_{i}^{∗})^{m}M ≠ 0}.  (6.18)

Note ε_{i}^{∗}(M) can be worked out just from knowledge of the character M: ε_{i}^{∗}(M) is the maximum k such that [L(i)^{⊛k} ⊛ …] appears in ch M.

Recalling the definition of the ideal I_{λ} generated by the element (4.6), Theorem 6.11(ii) has the following important corollary:

Corollary 6.13. Let λ ∈ P_{+} and M be an irreducible in Rep_{I} H_{n}. Then I_{λ}M = 0 if and only if ε_{i}^{∗}(M) ≤ ⟨h_{i}, λ⟩ for all i ∈ I.

§6-e. Functions φ_{i}. Fix λ ∈ P_{+} throughout this subsection. Let M be an irreducible H_{n}^{λ}-module. Recall from (5.10) that for i ∈ I,

ε_{i}(M) = max{m ≥ 0 | (ẽ_{i}^{λ})^{m}M ≠ 0},  (6.19)

since ẽ_{i}^{λ} is simply the restriction of ẽ_{i}. Analogously, we define

φ_{i}(M) = max{m ≥ 0 | (f̃_{i}^{λ})^{m}M ≠ 0}.  (6.20)

We will see shortly (Corollary 6.17) that φ_{i}(M) < ∞ always so that the definition makes sense. Note unlike ε_{i}(M), the integer φ_{i}(M) depends on the fixed choice of λ.

As in §5-d, 1_{λ} denotes the irreducible H_{0}^{λ}-module.

Lemma 6.14. ε_{i}(1_{λ}) = 0 and φ_{i}(1_{λ}) = ⟨h_{i}, λ⟩. Proof. The statement involving ε_{i} is obvious. For φ_{i}(1_{λ}), note that f̃_{i}^{m}1_{λ} = L(i^{m}) and

ε_{i}^{∗}(L(i^{m})) = m, ε_{j}^{∗}(L(i^{m})) = 0

for every j ≠ i. Hence by Corollary 6.13, pr^{λ}L(i^{m}) ≠ 0 if and only if m ≤ ⟨h_{i}, λ⟩. This implies that φ_{i}(1_{λ}) = ⟨h_{i}, λ⟩.

Lemma 6.15. Let i, j ∈ I with i ≠ j and M be an irreducible module in Rep_{I} H_{n}. Then, ε_{j}^{∗}(f̃_{i}^{m}M) ≤ ε_{j}^{∗}(M) for every m ≥ 0. Proof. Follows from the Shuffle Lemma.

Lemma 6.16. Let i, j ∈ I with i ≠ j. Let M be an irreducible module in Rep H_{n}^{λ} such that φ_{j}(M) > 0. Then, φ_{i}(f̃_{j}M) − ε_{i}(f̃_{j}M) ≤ φ_{i}(M) − ε_{i}(M) − ⟨h_{i}, α_{j}⟩.

Proof. Let ε = ε_{i}(M), φ = φ_{i}(M) and k = −⟨h_{i}, α_{j}⟩. By Lemma 5.23, there exist unique a, b ≥ 0 with a + b = k such that ε_{i}(f̃_{j}M) = ε − a. We need to show that φ_{i}(f̃_{j}M) ≤ φ + b, which follows if we can show that pr^{λ}f̃_{i}^{m}f̃_{j}M = 0 for all m > φ + b. We claim that

ε_{i}^{∗}(f̃_{i}^{m+b}f̃_{j}M) ≥ ε_{i}^{∗}(f̃_{i}^{m}M)

for all m ≥ 0. Given the claim, we know by the definition of φ, Corollary 6.13 and Lemma 6.15 that ε_{i}^{∗}(f̃_{i}^{m}M) > ⟨h_{i}, λ⟩ for all m > φ. So the claim implies that ε_{i}^{∗}(f̃_{i}^{m}f̃_{j}M) > ⟨h_{i}, λ⟩ for all m > φ + b, hence by Corollary 6.13 once more, pr^{λ}f̃_{i}^{m}f̃_{j}M = 0 as required.

To prove the claim, note that a ≤ ε, so b + ε ≥ k. Hence, Lemma 5.23(ii) shows that there is a surjection

ind_{n,m−a,a+b+1}^{n+m+b+1}M ⊛ L(i^{m−a}) ⊛ L(i^{a}ji^{b}) ↠ f̃_{i}^{m+b}f̃_{j}M.

By Lemma 5.19, res_{a,b+1}^{a+b+1}L(i^{a}ji^{b}) ≅ L(i^{a}) ⊛ L(ji^{b}). Hence by Frobenius reciprocity, there is a surjection

ind_{a,b+1}^{a+b+1}L(i^{a}) ⊛ L(ji^{b}) ↠ L(i^{a}ji^{b}).

Combining, we have proved existence of a surjection

ind_{n,m,b+1}^{n+m+b+1}M ⊛ L(i^{m}) ⊛ L(ji^{b}) ↠ f̃_{i}^{m+b}f̃_{j}M.

Hence by Frobenius reciprocity there is a non-zero map

(ind_{n,m}^{n+m}M ⊛ L(i^{m})) ⊛ L(ji^{b}) → res_{n+m,b+1}^{n+m+b+1}f̃_{i}^{m+b}f̃_{j}M.

Since the left hand module has irreducible cosocle f̃_{i}^{m}M ⊛ L(ji^{b}), we deduce that f̃_{i}^{m+b}f̃_{j}M has a constituent isomorphic to f̃_{i}^{m}M on restriction to the subalgebra H_{n+m} ⊆ H_{n+m+b+1}. This implies the claim.

Corollary 6.17. Let λ ∈ P_{+} and M be an irreducible H_{n}^{λ}-module with central character χ_{γ} for some γ ∈ Γ_{n}. Then, φ_{i}(M) − ε_{i}(M) ≤ ⟨h_{i}, λ − γ⟩. Proof. Proceed by induction on n, the case n = 0 being immediate by Lemma 6.14. For n > 0, we may write M = f̃_{j}N for some irreducible H_{n−1}^{λ}-module N with φ_{j}(N) > 0. By induction, φ_{i}(N) − ε_{i}(N) ≤ ⟨h_{i}, λ − γ + α_{j}⟩. The conclusion follows from Lemma 6.16.

§6-f. Alternative descriptions of φ_{i}. Keep λ ∈ P_{+} fixed. Now we wish to prove the analogue of Theorem 6.11 for the function φ_{i}. This is considerably more difficult to do. Let M be an irreducible H_{n}^{λ}-module. Recall that

f_{i}M = \underleftarrow{lim} pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i), ind_{i}M = \underleftarrow{lim} pr^{λ}ind_{n,1}^{n+1}M ⊠ R_{m}(i),

and that the inverse limits stabilize after finitely many terms. Define φ̃_{i}(M) to be the stabilization point of the limit, i.e. the least m ≥ 0 such that f_{i}M = pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i), or equivalently ind_{i}M = pr^{λ}ind_{n,1}^{n+1}M ⊠ R_{m}(i). The first lemma follows [G1, Theorem 9.15].

Lemma 6.18. Let M be an irreducible H_{n}^{λ}-module and i ∈ I. Then:

(i) [f_{i}M] = φ̃_{i}(M)[f̃_{i}M] + ∑ c_{a}[N_{a}] where the N_{a} are irreducible H_{n+1}^{λ}-modules with ε_{i}(N_{a}) < ε_{i}(f̃_{i}M);

(ii) End_{H_{n+1}^{λ}}(f_{i}M) ≃ End_{H_{n+1}^{λ}}(f̃_{i}M)^{⊕φ̃_{i}(M)} as vector superspaces.

Proof. (i) Take any m ≥ 1. Since pr^{λ} is right exact, the natural surjection L_{m}(i) ↠ L_{m−1}(i) and the obvious embedding L_{m}(i) ↪ L_{m+1}(i) (see §4-h) induce a commutative diagram

pr^{λ}ind_{n,1}^{n+1}M ⊛ L(i) ⟶^{α_{m}} pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i) ⟶^{β_{m}} pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m−1}(i) → 0
∥  ↓
pr^{λ}ind_{n,1}^{n+1}M ⊛ L(i) ⟶^{α_{m+1}} pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m+1}(i) ⟶^{β_{m+1}} pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i) → 0

where the rows are exact. Note if α_{m} = 0 then α_{m+1} = 0. It follows that if β_{m} is an isomorphism so is β_{m′} for every m′ ≥ m. So by definition of φ̃_{i}(M), the maps β_{1}, β_{2}, …, β_{φ̃_{i}(M)} are not isomorphisms but all other β_{m′}, m′ > φ̃_{i}(M) are isomorphisms.

Now to prove (i), we show by induction on m = 0, 1, …, φ̃_{i}(M) that

[pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i)] = m[f̃_{i}M] + lower terms,

where the lower terms are irreducible H_{n+1}^{λ}-modules N with ε_{i}(N) < ε_{i}(f̃_{i}M). This is vacuous if m = 0. For m > 0, β_{m} is not an isomorphism, so α_{m} ≠ 0. Hence, by Lemma 5.5, the image of α_{m} contains a copy of f̃_{i}M plus lower terms. Now the induction step is immediate.

(ii) Take m = φ̃_{i}(M). One easily shows using the explicit construction of L_{m}(i) in §4h that there is an even endomorphism θ : L_{m}(i) → L_{m}(i) of H_{1}-modules, such that the image of θ^{k} is ≃ L_{m−k}(i) for each 0 ≤ k ≤ m. Frobenius reciprocity induces superalgebra homomorphisms

End_{H_{1}}(L_{m}(i)) ↪ End_{H_{n,1}}(M ⊛ L_{m}(i)) ↪ End_{H_{n+1}}(ind_{n,1}^{n+1}M ⊛ L_{m}(i)).

So θ induces an even H_{n+1}-endomorphism θ̃ of ind_{n,1}^{n+1}M ⊛ L_{m}(i), such that the image of θ̃^{k} is ≃ ind_{n,1}^{n+1}M ⊛ L_{m−k}(i) for 0 ≤ k ≤ m. Now apply the right exact functor pr^{λ} to get an even H_{n+1}^{λ}-endomorphism

θ̂: pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i) → pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i)

induced by θ̃. Note θ̂^{m} = 0 and θ̂^{m−1} ≠ 0 because its image coincides with the image of the non-zero map α_{m} in the proof of (i). Hence, 1, θ̂, …, θ̂^{m−1} are linearly independent, even endomorphisms of f_{i}M. Now the proof of (ii) is completed in the same way as in the proof of Theorem 6.11(iii).

Corollary 6.19. Let M, N be irreducible H_{n}^{λ}-modules with M ≅ N̸. Then, for every i ∈ I, Hom_{H_{n+1}^{λ}}(f_{i}M, f_{i}N) = 0.

Proof. Repeat the argument in the proof of Corollary 6.12, but using φ̃_{i} and Lemma 6.18(i) in place of ε_{i} and Theorem 6.11(i).

The main thing now is to prove that φ̃_{i}(M) = φ_{i}(M). Note right away from the definitions that φ̃_{i}(M) = 0 if and only if φ_{i}(M) = 0.

Lemma 6.20. If M is an irreducible H_{n}^{λ}-module then

(φ̃_{0}(M) − ε_{0}(M)) + 2 ∑_{i=1}^{ℓ}(φ̃_{i}(M) − ε_{i}(M)) = ⟨c, λ⟩.

Proof. By Frobenius reciprocity and Theorem 3.9, we have that

End_{H_{n+1}^{λ}}(ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M) ≃ Hom_{H_{n}^{λ}}(M, res_{H_{n}^{λ}}^{H_{n+1}^{λ}}ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M)

≃ Hom_{H_{n}^{λ}}(M, M ⊕ ΠM)^{⊕⟨c,λ⟩} ⊕ Hom_{H_{n}^{λ}}(M, ind_{H_{n−1}^{λ}}^{H_{n}^{λ}}res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M)

≃ Hom_{H_{n}^{λ}}(M, M ⊕ ΠM)^{⊕⟨c,λ⟩} ⊕ End_{H_{n−1}^{λ}}(res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M).

Hence, by Schur’s lemma,

dim End_{H_{n+1}^{λ}}(ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M) − dim End_{H_{n−1}^{λ}}(res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M) = \{ \begin{matrix}2⟨c, λ⟩ & if M is of type M, \\ 4⟨c, λ⟩ & if M is of type Q. \end{matrix}

Now if M is of type M,

ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M ≃ f_{0}M ⊕ ⨁_{i=1}^{ℓ}(f_{i}M ⊕ Πf_{i}M), res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M ≃ e_{0}M ⊕ ⨁_{i=1}^{ℓ}(e_{i}M ⊕ Πe_{i}M).

by Lemmas 6.3, 6.4 and (6.6). Hence by Lemma 6.18(ii) and Theorem 6.11(iii),

dim End_{H_{n+1}^{λ}}(ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M) = 2φ̃_{0}(M) + 4 ∑_{i=1}^{ℓ} φ̃_{i}(M),

dim End_{H_{n−1}^{λ}}(res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M) = 2ε_{0}(M) + 4 ∑_{i=1}^{ℓ} ε_{i}(M),

and the conclusion follows in this case. The argument for M of type Q is similar.

Lemma 6.21. Let M be an irreducible H_{n}^{λ}-module and i ∈ I. Let c_{i} = 1 if i = 0, c_{i} = 2 otherwise. Then,

[res_{i}ind_{i}M : M] = 2c_{i}ε_{i}(f̃_{i}M)φ̃_{i}(M),  [ind_{i}res_{i}M : M] = 2c_{i}ε_{i}(M)φ̃_{i}(ẽ_{i}M),
soc res_{i}ind_{i}M ≃ (M ⊕ ΠM)^{⊕c_{i}φ̃_{i}(M)},  soc ind_{i}res_{i}M ≃ (M ⊕ ΠM)^{⊕c_{i}ε_{i}(M)}.

Proof. The statement about composition multiplicities follows from Theorem 6.11(i) and Lemma 6.18(i), taking into account how res_{i} and ind_{i} are related to e_{i} and f_{i} as explained in Lemmas 6.3 and 6.4. Now consider the statement about socles. We consider only res_{i}ind_{i}M, the other case being entirely similar but using results from §6-d instead. By adjointness, it suffices to be able to compute Hom_{H_{n+1}^{λ}}(ind_{i}N, ind_{i}M) for any irreducible H_{n}^{λ}-module N. But in view of Lemma 6.4, this can be computed from knowledge of Hom_{H_{n+1}^{λ}}(f_{i}N, f_{i}M) which is known by Corollary 6.19 (if M ≅ N̸) and Lemma 6.18(ii) (if M ≅ N). The details are similar to those in the proof of Lemma 6.20, so we omit them.

The proof of the next lemma is based on [V, Lemma 6.1].

Lemma 6.22. Let M be an irreducible H_{n}^{λ}-module and i ∈ I. There are maps

ind_{i}res_{i}M ⟶^{ψ} res_{i}ind_{i}M ⟶^{can} res_{i}ind_{i}M/soc res_{i}ind_{i}M,

whose composite is surjective. Proof. Let k = φ̃_{i}(M) and

π : ind_{n,1}^{n+1}M ⊠ R_{k}(i) ↠ pr^{λ}ind_{n,1}^{n+1}M ⊠ R_{k}(i) = ind_{i}M

be the quotient map. Let H′_{1} denote the subalgebra of H_{n} generated by X_{n}^{±1}, C_{n}, and set z = X_{n} + X_{n}^{−1} − q(i) (resp. X_{n} − 1 if i = 0). Recall from §4-h that viewed as an H′_{1}-module, we have that R_{k}(i) ≃ H′_{1}/(z^{k}). In particular, R_{k}(i) is a cyclic module generated by the image ˜1 of 1 ∈ H′_{1}.

We first observe that for any m ≥ ε_{i}(M) + φ̃_{i}(M), z^{m} annihilates the vector

π[T_{n} ⊗ (u ⊗ v)] ∈ ind_{i}M

for any u ∈ M, v ∈ R_{k}(i). This follows from the relations in H_{n+1}, e.g. in case i ≠ 0 one ultimately appeals to the facts that (X_{n} + X_{n}^{−1} − q(i))^{ε_{i}(M)} annihilates u (see Theorem 6.11(ii)) and (X_{n+1} + X_{n+1}^{−1} − q(i))^{φ̃_{i}(M)} annihilates v. It follows that the unique H_{n−1,1}-homomorphism (res_{i}M) ⊠ H′_{1} → res_{n−1,1}^{n}res_{i}ind_{i}M such that u ⊗ 1 ↦ π[T_{n} ⊗ (u ⊗ ˜1)] for each u ∈ res_{i}M ⊆ M factors to induce a well-defined H_{n−1,1}-module homorphism (res_{i}M) ⊠ R_{m}(i) → res_{n−1,1}^{n}res_{i}ind_{i}M. We then get from Frobenius reciprocity an induced map

ψ_{m} : ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{m}(i) → res_{i}ind_{i}M  (6.21)

for each m ≥ ε_{i}(M) + φ̃_{i}(M). Each ψ_{m} factors through the quotient pr^{λ}ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{m}(i), so we get an induced map

ψ : ind_{i}res_{i}M = \underleftarrow{lim} pr^{λ}ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{m}(i) → res_{i}ind_{i}M.

It remains to show that the composite of ψ with the canonical epimorphism from res_{i}ind_{i}M to res_{i}ind_{i}M/soc res_{i}ind_{i}M is surjective.

Let x = (n, n + 1) ∈ S_{n+1}. By Mackey Theorem there exists an exact sequence

0 → M ⊠ R_{k}(i) → res_{n,1}^{n+1}(ind_{n,1}^{n+1}M ⊠ R_{k}(i)) → ind_{n−1,1,1}^{n,1}^{x}((res_{n−1,1}^{n}M) ⊠ R_{k}(i)) → 0.

In other words, there is an H_{n,1}-isomorphism

ind_{n−1,1,1}^{n,1}^{x}((res_{n−1,1}^{n}M) ⊠ R_{k}(i)) ⟶^{∼} res_{n,1}^{n+1}(ind_{n,1}^{n+1}M ⊠ R_{k}(i))/(M ⊠ R_{k}(i)),
h ⊗ (u ⊗ v) ↦ hT_{n} ⊗ u ⊗ v + M ⊠ R_{k}(i)

for h ∈ H_{n}, u ∈ M, v ∈ R_{k}(i), where M ⊠ R_{k}(i) is embedded into res_{n,1}^{n+1}(ind_{n,1}^{n+1}M ⊠ R_{k}(i)) as 1 ⊗ M ⊗ R_{k}(i). Recall from (4.27) that dim R_{k}(i) = 2kc_{i} where c_{i} is as in Lemma 6.21. Hence, applying Lemma 6.21,

res_{n}^{n,1}M ⊠ R_{k}(i) ≃ (M ⊕ ΠM)^{kc_{i}} ≃ soc res_{i}ind_{i}M.

So applying the exact functor res_{i} = res_{n}^{n,1} ∘ Δ_{i} to the isomorphism above we get an isomorphism

ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{k}(i) ⟶^{∼} res_{i}(ind_{n,1}^{n+1}M ⊠ R_{k}(i))/soc res_{i}ind_{i}M,
h ⊗ u ⊗ v ↦ hT_{n} ⊗ u ⊗ v + soc res_{i}ind_{i}M.

It follows that there is a surjection

θ : ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{k}(i) ↠ res_{i}ind_{i}M/soc res_{i}ind_{i}M.

such that the diagram

ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{m}(i) ⟶^{ψ_{m}}  res_{i}ind_{i}M
↓  ↓^{can}
ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{k}(i) ⟶^{θ} res_{i}ind_{i}M/soc res_{i}ind_{i}M

commutes for all m ≥ ε_{i}(M) + φ̃_{i}(M), where ψ_{m} is the map from (6.21) and the left hand arrow is the natural surjection. Now surjectivity of θ immediately implies the desired surjectivity of the composite.

Lemma 6.23. Let M be an irreducible H_{n}^{λ}-module with ε_{i}(M) > 0. Then,

φ̃_{i}(ẽ_{i}M) = φ̃_{i}(M) + 1.

Proof. Let us first show that

φ̃_{i}(ẽ_{i}M) ≥ φ̃_{i}(M) + 1.  (6.22)

Recall φ_{i}(M) = 0 if and only if φ̃_{i}(M) = 0. Suppose first that φ_{i}(M) = 0, when φ_{i}(ẽ_{i}M) ≠ 0. Then, φ̃_{i}(M) = 0 and φ̃_{i}(ẽ_{i}M) ≠ 0, so the conclusion certainly holds in this case. So we may assume that φ_{i}(M) > 0, hence φ̃_{i}(M) ≠ 0. Note by Lemma 6.21,

[res_{i}ind_{i}M/soc res_{i}ind_{i}M : M] = 2c_{i}ε_{i}(f̃_{i}M)φ̃_{i}(M) − 2c_{i}φ̃_{i}(M) = 2c_{i}ε_{i}(M)φ̃_{i}(M) ≠ 0.

In particular, the map ψ in Lemma 6.22 is non-zero. Now Lemma 6.22 implies that the multiplicity of M in im ψ is strictly greater than 2c_{i}ε_{i}(M)φ̃_{i}(M), since at least one composition factor of soc im ψ ⊆ soc res_{i}ind_{i}M must be sent to zero on composing with the second map can. Using another part of Lemma 6.21, this shows that

2c_{i}ε_{i}(M)φ̃_{i}(ẽ_{i}M) > 2c_{i}ε_{i}(M)φ̃_{i}(M)

and (6.22) follows.

Now using (6.22) and Lemma 6.21, we see that in the Grothendieck group,

[res_{i}ind_{i}M − ind_{i}res_{i}M : M] ≤ 2c_{i}(φ̃_{i}(M) − ε_{i}(M)),

with equality if and only if equality holds in (6.22). By central character considerations, for i ≠ j, [res_{i}ind_{j}M : M] = [ind_{j}res_{i}M : M] = 0. So using (6.6) we deduce that

[res_{H_{n}^{λ}}^{H_{n+1}^{λ}}ind_{H_{n}^{λ}}^{H_{n+1}^{λ}}M − ind_{H_{n−1}^{λ}}^{H_{n}^{λ}}res_{H_{n−1}^{λ}}^{H_{n}^{λ}}M : M] ≤ 2(φ̃_{0}(M) − ε_{0}(M)) + 4 ∑_{i=1}^{ℓ}(φ̃_{i}(M) − ε_{i}(M))

with equality if and only if equality holds in (6.22) for all i ∈ I. Now Lemma 6.20 shows that the right hand side equals 2⟨c, λ⟩, which does indeed equal the left hand side thanks to Theorem 3.9.

Corollary 6.24. For any irreducible H_{n}^{λ}-module M, φ_{i}(M) = φ̃_{i}(M). Proof. We proceed by induction on φ_{i}(M), the conclusion being known already in case φ_{i}(M) = 0. For the induction step, take an irreducible H_{n}^{λ}-module N with φ_{i}(N) > 0, so N = ẽ_{i}M where M = f̃_{i}N is an irreducible H_{n+1}^{λ}-module with ε_{i}(M) > 0, φ_{i}(M) < φ_{i}(N). Then by Lemma 6.23 and the induction hypothesis,

φ̃_{i}(N) = φ̃_{i}(ẽ_{i}M) = φ̃_{i}(M) + 1 = φ_{i}(M) + 1 = φ_{i}(ẽ_{i}M) = φ_{i}(N).

This completes the induction step.

As a first consequence, we can improve Corollary 6.17:

Lemma 6.25. Let M be an irreducible H_{n}^{λ}-module of central character χ_{γ} for γ ∈ Γ_{n}. Then, φ_{i}(M) − ε_{i}(M) = ⟨h_{i}, λ − γ⟩.

Proof. In view of Corollary 6.17, it suffices to show that

(φ_{0}(M) − ε_{0}(M)) + 2 ∑_{i=1}^{ℓ}(φ_{i}(M) − ε_{i}(M)) = ⟨c, λ⟩.

But this is immediate from Lemma 6.20 and Corollary 6.24.

We are finally ready to assemble the results of the subsection to obtain the full analogue of Theorem 6.11 for φ_{i}:

Theorem 6.26. Let i ∈ I and M be an irreducible H_{n}^{λ}-module. Then,

(i) [f_{i}M] = φ_{i}(M)[f̃_{i}M] + ∑ c_{a}[N_{a}] where the N_{a} are irreducibles with φ_{i}(N_{a}) < φ_{i}(f̃_{i}M);

(ii) φ_{i}(M) is the least m ≥ 0 such that f_{i}M = pr^{λ}ind_{n,1}^{n+1}M ⊛ L_{m}(i); (iii) End_{H_{n+1}}(f_{i}M) ≃ End_{H_{n+1}}(f̃_{i}M)^{⊕φ_{i}(M)} as vector superspaces.

Proof. (i) Since φ_{i}(M) = φ̃_{i}(M) by Corollary 6.24, we know by Lemma 6.18(i) that

[f_{i}M] = φ_{i}(M)[f̃_{i}M] + ∑ c_{a}[N_{a}]

where the N_{a} are irreducibles with ε_{i}(N_{a}) < ε_{i}(f̃_{i}M). Suppose that M ∈ Rep_{γ} H_{n}^{λ}, for γ ∈ Γ_{n}. Then f̃_{i}M and each N_{a} have central character χ_{γ+α_{i}}, since they are all composition factors of ind_{i}M. So by Lemma 6.25,

φ_{i}(N_{a}) = ⟨h_{i}, λ − γ − α_{i}⟩ + ε_{i}(N_{a}), φ_{i}(f̃_{i}M) = ⟨h_{i}, λ − γ − α_{i}⟩ + ε_{i}(f̃_{i}M).

It follows that φ_{i}(N_{a}) < φ_{i}(f̃_{i}N_{a}) too.

(ii) This is just the definition of φ̃_{i}(M) combined with Corollary 6.24.

(iii) This follows from Lemma 6.18(ii).

7. Bernstein-Macdonald Hopf algebra

§7-a. Grothendieck groups revisited. Let us now write

K(∞) = ⨁_{n≥0} K(Rep_{I} H_{n})  (7.1)

for the sum over all n of the Grothendieck groups of the categories Rep_{I} H_{n}. Also, write

K(∞)_{ℚ} = ℚ ⊗_{ℤ} K(∞),  (7.2)

extending scalars. Thus K(∞) is a free ℤ-module with canonical basis given by B(∞), the isomorphism classes of irreducible modules, and K(∞)_{ℚ} is the ℚ-vector space on basis B(∞). We will always view K(∞) as a lattice in K(∞)_{ℚ}.

We let K(∞)^{∗} denote the restricted dual of K(∞), namely, the set of functions f : K(∞) → ℤ such that f vanishes on all but finitely many elements of B(∞). Thus K(∞)^{∗} is also a free ℤ-module, with canonical basis

{δ_{M} | [M] ∈ B(∞)}

dual to the basis B(∞) of K(∞), i.e. δ_{M}([M]) = 1, δ_{M}([N]) = 0 for [N] ∈ B(∞) with N ≅ M̸. Note for an arbitrary N ∈ Rep_{I} H_{n}, δ_{M}([N]) simply computes the composition multiplicity [N : M] of the irreducible module M as a composition factor of N. Finally, we write B(∞)_{ℚ}^{∗} := ℚ ⊗_{ℤ} B(∞)^{∗}, which can be identified with the restricted dual of B(∞)_{ℚ}.

Entirely similar definitions can be made for each λ ∈ P_{+}:

K(λ) = ⨁_{n≥0} K(Rep_{I} H_{n}^{λ})  (7.3)

denotes the Grothendieck groups of the categories Rep H_{n}^{λ} for all n. Again, K(λ) is a free ℤ-module on the basis B(λ) of isomorphism classes of irreducible modules. Moreover, infl^{λ} induces a canonical embedding

infl^{λ} : K(λ) ↪ K(∞).

We will generally identify K(λ) with its image under this embedding. We also define K(λ)^{∗} and K(λ)_{ℚ} = ℚ ⊗_{ℤ} K(λ) as above.

Recall the operators e_{i} and more generally the divided power operators e_{i}^{(r)} for r ≥ 1, defined on irreducible modules in res_{I}H_{n} in (6.8) and (6.11) respectively. These induce linear maps

e_{i}^{(r)} : K(∞) → K(∞)  (7.4)

for each r ≥ 1. Similarly, the operators e_{i}^{(r)} and f_{i}^{(r)} from (6.12) and (6.13) respectively induce maps

e_{i}^{(r)}, f_{i}^{(r)} : K(λ) → K(λ).  (7.5)

Recall by Lemma 6.9 that

e_{i}^{r} = (r!)e_{i}^{(r)},  f_{i}^{r} = (r!)f_{i}^{(r)}.  (7.6)

Extending scalars, the maps e_{i}^{(r)}, f_{i}^{(r)} induce linear maps on K(∞)_{ℚ} and K(λ)_{ℚ} too.

§7-b. Hopf algebra structure. Now we wish to give K(∞) the structure of a graded Hopf algebra over ℤ. To do this, recall the canonical isomorphism

K(Rep_{I} H_{m}) ⊗_{ℤ} K(Rep_{I} H_{n}) → K(Rep_{I} H_{m,n})  (7.7)

from (2.4), for each m, n ≥ 0. The exact functor ind_{m,n}^{m+n} induces a well-defined map

ind_{m,n}^{m+n} : K(Rep_{I} H_{m,n}) → K(Rep_{I} H_{m+n}).

Composing with the isomorphism (7.7) and taking the direct sum over all m, n ≥ 0, we obtain a homogeneous map

⋄ : K(∞) ⊗_{ℤ} K(∞) → K(∞).  (7.8)

By transitivity of induction, this makes K(∞) into an associative graded ℤ-algebra. Moreover, there is a unit

ι : ℤ → K(∞)  (7.9)

mapping 1 to the identity module 1 of H_{0}. Finally, since the duality τ induces the identity map at the level of Grothendieck groups, Theorem 2.14 implies that the multiplication ⋄ is commutative (in the usual unsigned sense).

Now consider how to define the comultiplication. The exact functor res_{n_{1},n_{2}}^{n} induce a map

res_{n_{1},n_{2}}^{n} : K(Rep_{I} H_{n}) → K(Rep_{I} H_{n_{1},n_{2}}).

On composing with the isomorphism (7.7), we obtain maps

Δ_{n_{1},n_{2}}^{n} :K(Rep_{I} H_{n}) → K(Rep_{I} H_{n_{1}}) ⊗_{ℤ} K(Rep_{I} H_{n_{2}}),  (7.10)
Δ^{n} = ∑_{n_{1}+n_{2}=n} Δ_{n_{1},n_{2}}^{n} :K(Rep_{I} H_{n}) → ⨁_{n_{1}+n_{2}=n} K(Rep_{I} H_{n_{1}}) ⊗_{ℤ} K(Rep_{I} H_{n_{2}}). (7.11)

Now taking the direct sum over all n ≥ 0 gives a homomogeneous map

Δ : K(∞) → K(∞) ⊗_{ℤ} K(∞).  (7.12)

Transitivity of restriction implies that Δ is coassociative, while the homogeneous projection onto K(Rep_{I} H_{0}) ≅ ℤ gives a counit

ε : K(∞) → ℤ.  (7.13)

Thus K(∞) is also a graded coalgebra over ℤ. Now finally:

Theorem 7.1. (K(∞), ⋄, Δ, ι, ε) is a commutative, graded Hopf algebra over ℤ. Proof. It just remains to check that Δ is an algebra homomorphism, which follows using the Mackey Theorem. Note in checking the details, one needs to use Lemma 5.14 to take the definition of ⊛ into account correctly.

We record the following lemma explaining how to compute the action of e_{i} on K(∞) explicitly in terms of Δ:

Lemma 7.2. Let M be a module in Rep_{I} H_{n}. Write

Δ_{n−1,1}^{n}[M] = ∑_{a}[M_{a}] ⊗ [N_{a}]

for irreducible H_{n−1}-modules M_{a} and irreducible H_{1}-modules N_{a}. Then,

e_{i}[M] = ∑_{a with N_{a}≅L(i)}[M_{a}].

Proof. This is immediate from Lemma 6.3.

Lemma 7.3. The operators e_{i} : K(∞) → K(∞) satisfy the Serre relations, i.e.

e_{i}e_{j} = e_{j}e_{i}  if |i − j| > 1;
e_{i}^{2}e_{j} + e_{j}e_{i}^{2} = 2e_{i}e_{j}e_{i}  if |i − j| = 1, i ≠ 0, j ≠ ℓ;
e_{0}^{3}e_{1} + 3e_{0}e_{1}e_{0}^{2} = 3e_{0}^{2}e_{1}e_{0} + e_{1}e_{0}^{3}  if ℓ ≠ 1;
e_{ℓ−1}^{3}e_{ℓ} + 3e_{ℓ−1}e_{ℓ}e_{ℓ−1}^{2} = 3e_{ℓ−1}^{2}e_{ℓ}e_{ℓ−1} + e_{ℓ}e_{ℓ−1}^{3}  if ℓ ≠ 1;
e_{0}^{5}e_{1} + 5e_{0}e_{1}e_{0}^{4} + 10e_{0}^{3}e_{1}e_{0}^{2} = 10e_{0}^{2}e_{1}e_{0}^{3} + 5e_{0}^{4}e_{1}e_{0} + e_{1}e_{0}^{5}  if ℓ = 1,

for i, j ∈ I. Proof. In view of Lemma 7.2 and coassociativity of Δ, this reduces to checking it on irreducible H_{n}-modules for n = 2, 3, 4, 4, 6 respectively. For this, the character information in Lemmas 5.19 and 5.20 is sufficient.

Now consider K(λ) for λ ∈ P_{+}. This has a natural structure as K(∞)-comodule: viewing K(λ) as a subset of K(∞), the comodule structure map is the restriction

Δ^{λ} : K(λ) → K(λ) ⊗_{ℤ} K(∞)

of Δ. In other words, each K(λ) is a subcomodule of the right regular K(∞)-comodule.

One checks from the definitions that the dual maps to ⋄, Δ, ι, ε induce on K(∞)^{∗} the structure of a cocommutative graded Hopf algebra. ¿From this point of view, each K(λ) is a left K(∞)^{∗}-module in the natural way: f ∈ K(∞)^{∗} acts on the left on K(λ) as the map (id ¯⊗f) ∘ Δ^{λ}. Similarly, K(∞) is itself a left K(∞)^{∗}-module, indeed in this case the action is even faithful.

Lemma 7.4. The operator e_{i}^{(r)} acts on K(∞) (resp. K(λ) for any λ ∈ P_{+}) in the same way as the basis element δ_{L(i^{r})} of K(∞)^{∗}. Proof. Let M be an irreducible module in Rep_{I} H_{n} or Rep H_{n}^{λ}. Let

Δ_{n−r,r}[M] = ∑_{a}[M_{a}] ⊗ [N_{a}]

for irreducible H_{n−r}^{λ}-modules M_{a} and irreducible H_{r}-modules N_{a}. By the definition of the action of K(∞)^{∗} on K(λ), it follows that

δ_{L(i^{r})}[M] = ∑_{a with N_{a}≅L(i^{r})}[M_{a}].

Hence, since [res_{1,...,1}^{r}L(i^{r})] = (r!)[L(i)^{⊛r}], we get that δ_{L(i)}^{r} acts in the same way as (r!)δ_{L(i^{r})}. So in view of (7.6), it just remains to check that δ_{L(i)} acts in the same way as e_{i}, which follows by Lemma 7.2.

Lemma 7.5. There is a unique homomorphism π : U_{ℤ}^{+} → K(∞)^{∗} of graded Hopf algebras such that e_{i}^{(r)} ↦ δ_{L(i^{r})} for each i ∈ I and r ≥ 1. Proof. Since K(∞) is a faithful K(∞)^{∗}-module, (7.6) and Lemmas 7.3 and 7.4 imply that the operators δ_{L(i^{r})} satisfy the same relations as the generators e_{i}^{(r)} of U_{ℤ}^{+}. This implies existence of a unique such algebra homomorphism. The fact that π is a coalgebra map follows because

Δ(δ_{L(i)}) = δ_{L(i)} ⊗ 1 + 1 ⊗ δ_{L(i)}

by the definition of the comultiplication on K(∞)^{∗}.

§7-c. Shapovalov form. Now focus on a fixed λ ∈ P_{+}. For an H_{n}^{λ}-module M, we let P_{M} denote its projective cover in the category H_{n}^{λ}-mod. Since H_{n}^{λ} is a finite dimensional superalgebra, we can identify

K(λ)^{∗} = ⨁_{n≥0} K(Proj H_{n}^{λ})  (7.14)

so that the basis element δ_{M} corresponds to the isomorphism class [P_{M}] for each irreducible H_{n}^{λ}-module M and each n ≥ 0. Moreover, under this identification, the canonical pairing

(., .) : K(λ)^{∗} × K(λ) → ℤ  (7.15)

satisfies

([P_{M}], [N]) = \{ \begin{matrix}dim Hom_{H_{n}^{λ}}(P_{M}, N) & if M is of type M, \\ \frac{1}{2} dim Hom_{H_{n}^{λ}}(P_{M}, N) & if M is of type Q, \end{matrix}  (7.16)

for H_{n}^{λ}-modules M, N with M irreducible (since the right hand side clearly computes the composition multiplicity [N : M]).

There is a homogeneous map

ω : K(λ)^{∗} → K(λ)  (7.17)

induced by the natural maps K(Proj H_{n}^{λ}) → K(Rep H_{n}^{λ}) for each n. We warn the reader that we do not yet know that ω is injective.

As explained at the end of §6-c, we can define an action of e_{i}^{(r)} and f_{i}^{(r)} on the projective indecomposable modules, hence on K(λ)^{∗}. We know by Lemma 6.10 that (7.6) holds for the operations on K(λ)^{∗} as well as on K(λ). Also, the actions of e_{i}^{(r)} and f_{i}^{(r)} commute with ω by (6.14).

Lemma 7.6. The operators e_{i}, f_{i} on K(λ)^{∗} and K(λ) satisfy (e_{i}x, y) = (x, f_{i}y), (f_{i}x, y) = (x, e_{i}y) for each x ∈ K(λ)^{∗} and y ∈ K(λ). Proof. Let M be an irreducible H_{n}^{λ}-module and N be an irreducible H_{n+1}^{λ}-module. We check that (f_{i}[P_{M}], [N]) = ([P_{M}], e_{i}[N]) in the special case that i = 0, M is of type Q and N is of type M. In this case, Lemmas 6.4, 6.1(i) and 6.3,

(f_{i}[P_{M}], [N]) = \frac{1}{2}(ind_{i}[P_{M}], [N]) = \frac{1}{2} dim Hom_{H_{n+1}^{λ}}(ind_{i}P_{M}, N)
= \frac{1}{2} dim Hom_{H_{n}^{λ}}(P_{M}, res_{i}N) = ([P_{M}], res_{i}[N]) = ([P_{M}], e_{i}[N]).

All the other situations that need to be considered follow similarly.

Corollary 7.7. Suppose

e_{i}^{(r)}[M] = ∑_{[N]∈B(λ)} a_{M,N}[N], f_{i}^{(r)}[M] = ∑_{[N]∈B(λ)} b_{M,N}[N].

for [M] ∈ B(λ). Then,

e_{i}^{(r)}[P_{N}] = ∑_{[M]∈B(λ)} b_{M,N}[P_{M}], f_{i}^{(r)}[P_{N}] = ∑_{[M]∈B(λ)} a_{M,N}[P_{M}]

for [N] ∈ B(λ).

Using this, the next lemma is an easy consequence of Theorem 6.26(i), see [G1, Lemma 11.2] for the detailed proof.

Lemma 7.8. Let M be an irreducible H_{n}^{λ}-module, set ε = ε_{i}(M), φ = φ_{i}(M). Then, for any m ≥ 0,

e_{i}^{(m)}[P_{M}] = ∑_{N with ε_{i}(N)≥m} a_{N}[P_{ẽ_{i}^{m}N}]

for coefficients a_{N} ∈ ℤ_{≥0}. Moreover, in case m = ε,

e_{i}^{(ε)}[P_{M}] = \binom{ε + φ}{ε}[P_{ẽ_{i}^{ε}M}] + ∑_{N with ε_{i}(N)>ε} a_{N}[P_{ẽ_{i}^{ε}N}].

We also need:

Theorem 7.9. Given an irreducible H_{n}^{λ}-module M, [P_{M}] ∈ K(Proj H_{n}^{λ}) can be written as an integral linear combination of terms of the form f_{i_{1}}^{(r_{1})} … f_{i_{a}}^{(r_{a})}[1_{λ}].

Proof. Proceed by induction on n, the conclusion being trivial for n = 0. So suppose n > 0 and that the result is true for all smaller n. Suppose for a contradiction that we can find an irreducible H_{n}^{λ}-module M for which the result does not hold. Pick i with ε := ε_{i}(M) > 0. Since there are only finitely many irreducible H_{n}^{λ}-modules, we may assume by the choice of M that the result holds for all irreducible H_{n}^{λ}-modules L with ε_{i}(L) > ε. Write

f_{i}^{(ε)}[P_{ẽ_{i}^{ε}M}] = ∑_{[L]∈Irr H_{n}^{λ}} a_{L}[P_{L}]

for coefficients a_{L} ∈ ℤ. By Corollary 7.7, a_{L} = [e_{i}^{(ε)}L : ẽ_{i}^{ε}M]. In particular, a_{L} = 0 unless ε_{i}(L) ≥ ε. Moreover, if a_{L} ≠ 0 for L with ε_{i}(L) = ε, then by Theorem 6.11(i), we have that e_{i}^{(ε)}L ≅ ẽ_{i}^{ε}L ≅ ẽ_{i}^{ε}M, whence L ≅ M and a_{M} = 1. This shows:

[P_{M}] = f_{i}^{(ε)}[P_{ẽ_{i}^{ε}M}] − ∑_{L with ε_{i}(L)>ε} a_{L}[P_{L}]  (7.18)

for some a_{L} ∈ ℤ. But the inductive hypothesis and choice of M ensure that all terms on the right hand side are integral linear combinations of terms f_{i_{1}}^{(r_{1})} … f_{i_{a}}^{(r_{a})}[1_{λ}], hence the same is true for [P_{M}], a contradiction.

The next two theorems are fundamental, so we include the proofs even though they are identical to the argument in [G1, Theorem 11.1].

Theorem 7.10. The map ω : K(λ)^{∗} → K(λ) from (7.17) is injective. Proof. We show by induction on n that the map ω : K(Proj H_{n}^{λ}) → K(Rep H_{n}^{λ}) is injective. This is clear if n = 0, so assume n > 0 and the result has been proved for all smaller n. Suppose we have a relation

∑_{[M]∈Irr H_{n}^{λ}} a_{M}ω([P_{M}]) = 0

with not all coefficients a_{M} being zero. We may choose i ∈ I and an irreducible module M such that a_{M} ≠ 0, ε := ε_{i}(M) > 0 and a_{N} = 0 for all N with ε_{i}(N) > ε.

Apply e_{i}^{(ε)} to the sum. By Lemma 7.8, we get

∑_{N with ε_{i}(N)=ε} \binom{ε + φ_{i}(N)}{ε}a_{N}ω([P_{ẽ_{i}^{ε}N}]) + X = 0

where X is a sum of terms of the form ω([P_{ẽ_{i}^{ε}L}]) with ε_{i}(L) > ε. Now the inductive hypothesis shows that X = 0 and that a_{N} = 0 for each N with ε_{i}(N) = ε. In particular, a_{M} = 0, a contradiction.

In view of Theorem 7.10, we may identify K(λ)^{∗} with its image under ω, so K(λ)^{∗} ⊆ K(λ) are two different lattices in K(λ)_{ℚ}: on tensoring with ℚ they become equal. Extending scalars, the pairing (7.15) induces a bilinear form

(., .) : K(λ)_{ℚ} × K(λ)_{ℚ} → ℚ  (7.19)

with respect to which the operators e_{i} and f_{i} are adjoint.

Theorem 7.11. The form (., .) : K(λ)_{ℚ} × K(λ)_{ℚ} → ℚ is symmetric and non-degenerate. Proof. It is non-degenerate by construction, being induced by the pairing (., .) from (7.15). So we just need to check that it is symmetric. Proceed by induction on n to show that (., .) : K(Rep H_{n}^{λ})_{ℚ} × K(Rep H_{n}^{λ})_{ℚ} → ℚ is symmetric. In view of Theorem 7.9, any element of K(Rep H_{n}^{λ})_{ℚ} can be written as f_{i}x for x ∈ K(Rep H_{n−1}^{λ})_{ℚ}. Then for any other y ∈ K(Rep H_{n}^{λ})_{ℚ}, we have that

(f_{i}x, y) = (x, e_{i}y) = (e_{i}y, x) = (y, f_{i}x)

by the induction hypothesis.

§7-d. Chevalley relations. Continue working with a fixed λ ∈ P_{+}. We turn now to considering the relations satisfied by the operators e_{i}, f_{i} on K(λ).

Lemma 7.12. The operators e_{i}, f_{i} : K(λ) → K(λ) satisfy the Serre relations (4.4). Proof. We know the e_{i} satisfy the Serre relations on all of K(∞) by Lemma 7.3, so they certainly satisfy the Serre relations on restriction to K(λ). Moreover, e_{i} and f_{i} are adjoint operators for the bilinear form (., .) according to Lemma 7.6, and this form is non-degenerate by Theorem 7.11. The lemma follows.

Now we consider relations between the e_{i} and f_{j}. For i ∈ I and an irreducible H_{n}^{λ}-module M with central character χ_{γ} for γ ∈ Γ_{n}, define

h_{i}[M] = ⟨h_{i}, λ − γ⟩[M].  (7.20)

Recall according to Lemma 6.25 that we have equivalently that

h_{i}[M] = (φ_{i}(M) − ε_{i}(M))[M].  (7.21)

More generally, define

\binom{h_{i}}{r} : K(λ) → K(λ),  [M] ↦ \binom{φ_{i}(M) − ε_{i}(M)}{r}[M]  (7.22)

where \binom{m}{r} denotes m(m−1) … (m−r +1)/(r!). Extending linearly, each \binom{h_{i}}{r} can be viewed as a diagonal linear operator K(λ) → K(λ). The definition (7.20) implies immediately that:

Lemma 7.13. As operators on K(λ), [h_{i}, e_{j}] = ⟨h_{i}, α_{j}⟩e_{j} and [h_{i}, f_{j}] = −⟨h_{i}, α_{j}⟩f_{j} for all i, j ∈ I.

For the next lemma, we follow [G1, Proposition 12.5].

Lemma 7.14. As operators on K(λ), the relation [e_{i}, f_{j}] = δ_{i,j}h_{i} holds for each i, j ∈ I. Proof. Let M be an irreducible H_{n}^{λ}-module. It follows immediately from Theorems 6.11(i) and 6.26(i) (together with central character considerations in case i ≠ j) that [M] appears in e_{i}f_{j}[M] − f_{j}e_{i}[M] with multiplicity δ_{i,j}(φ_{i}(M) − ε_{i}(M)). Therefore, it suffices simply to show that e_{i}f_{j}[M] − f_{j}e_{i}[M] is a multiple of [M]. Let us show equivalently that

[res_{i}ind_{j}M − ind_{j}res_{i}M]

is a multiple of [M].

For m ≫ 0, we have a surjection ind_{n,1}^{n+1}M ⊠ R_{m}(j) ↠ ind_{j}M. Apply pr^{λ} ∘ res_{i} to get a surjection

pr^{λ}res_{i}ind_{n,1}^{n+1}M ⊠ R_{m}(j) ↠ res_{i}ind_{j}M.  (7.23)

By the Mackey Theorem and (4.28), there is an exact sequence

0 → (M ⊕ ΠM)^{⊕δ_{i,j}.mc_{j}} ⟶ res_{i}ind_{n,1}^{n+1}M ⊠ R_{m}(j) ⟶ ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{m}(j) → 0, where c_{j} is as in Lemma 6.21. For sufficiently large m, pr^{λ}ind_{n−1,1}^{n}(res_{i}M) ⊠ R_{m}(j) = ind_{j}res_{i}M. So on applying the right exact functor pr^{λ} and using the irreducibility of M, this implies that there is an exact sequence

0 ⟶ M^{⊕m_{1}} ⊕ ΠM^{⊕m_{2}} ⟶ pr^{λ}res_{i}ind_{n,1}^{n+1}M ⊠ R_{m}(j) ⟶ ind_{j}res_{i}M ⟶ 0,  (7.24)

for some m_{1}, m_{2}. Now let N be any irreducible H_{n}^{λ}-module with N ≅ M̸. Combining (7.23) and (7.24) shows that

[ind_{j}res_{i}M − res_{i}ind_{j}M : N] ≥ 0  (7.25)

Now summing over all i, j and using (6.6) gives that [ind res M − res ind M : N] ≥ 0. But Theorem 3.9 shows that equality holds here, hence it must hold in (7.25) for all i, j ∈ I. This completes the proof.

To summarize, we have shown in (7.6), Lemmas 7.12, 7.13 and 7.14 that:

Theorem 7.15. The action of the operators e_{i}, f_{i}, h_{i} on K(λ) satisfy the Chevalley relations (4.2), (4.3) and (4.4). Hence, the actions of e_{i}^{(r)}, f_{i}^{(r)} and \binom{h_{i}}{r} for all i ∈ I, r ≥ 1 make K(λ)_{ℚ} into a U_{ℚ}-module so that K(λ)^{∗}, K(λ) are U_{ℤ}-submodules.

§7-e. Identification of K(∞)^{∗}, K(λ)^{∗} and K(λ). Now we can prove Theorems A and B stated in the introduction.

Theorem 7.16. For any λ ∈ P_{+},

(i) K(λ)_{ℚ} is precisely the integrable highest weight U_{ℚ}-module of highest weight λ, with highest weight vector [1_{λ}];

(ii) the bilinear form (., .) from (7.19) on the highest weight module K(λ)_{ℚ} coincides

with the usual Shapovalov form satisfying ([1_{λ}], [1_{λ}]) = 1; (iii) K(λ)^{∗} ⊂ K(λ) are integral forms of K(λ)_{ℚ} containing [1_{λ}], with K(λ)^{∗} being the

minimal lattice U_{ℤ}^{−}[1_{λ}] and K(λ) being its dual under the Shapovalov form. Proof. It makes sense to think of K(λ)_{ℚ} as a U_{ℚ}-module according to Theorem 7.15. The actions of e_{i} and f_{i} are locally nilpotent by Theorems 6.11(i) and 6.26(i). The action of h_{i} is diagonal by definition. Hence, K(λ)_{ℚ} is an integrable module. Clearly [1_{λ}] is a highest weight vector of highest weight λ. Moreover, K(λ)_{ℚ} = U_{ℚ}^{−}[1_{λ}] by Theorem 7.9. This completes the proof of (i), and (ii) follows immediately from Lemma 7.6. For (iii), we know already that K(λ)^{∗} ⊂ K(λ) are dual lattices of K(λ)_{ℚ} which are invariant under U_{ℤ}. Moreover, Theorem 7.9 again shows K(λ)^{∗} = U_{ℤ}^{−}[1_{λ}].

Theorem 7.17. The map π : U_{ℤ}^{+} → K(∞)^{∗} constructed in Lemma 7.5 is an isomorphism.

Proof. Note by Lemma 7.4 that the action of the e_{i}^{(r)} ∈ U_{ℤ}^{+} on K(∞), hence on each K(λ), factors through the map π. So if x ∈ ker π, we have by the previous theorem that x acts as zero on all integrable highest weight modules K(λ), λ ∈ P_{+}. Hence x = 0 and π is injective. To prove surjectivity, take x ∈ K(∞). It suffices to show that (π(u), x) = 0 for all u ∈ U_{ℤ}^{+} implies that x = 0. Note

(π(u), x) = (π(1)π(u), x) = (π(1), π(u)x) = (π(1), ux)

where the second equality follows because the right regular action of K(∞)^{∗} on itself is precisely the dual action to the left action of K(∞)^{∗} on K(∞), and the third equality follows from Lemma 7.4. Hence, if (π(u), x) = 0 for all u ∈ U_{ℤ}^{+}, we have that (π(1), ux) = 0 for all u ∈ U_{ℤ}^{+}. Now choose λ ∈ P_{+} sufficiently large so that in fact x ∈ K(λ) ⊂ K(∞).

Then, it follows that ([1_{λ}], ux) = 0 for all u ∈ U_{ℤ}^{+}, where (., .) now is canonical pairing between K(λ)^{∗} and K(λ). Hence by Lemma 7.6, (v[1_{λ}], x) = 0 for all v ∈ U_{ℤ}^{−}. But then Theorem 7.9 implies that x = 0.

8. Identification of the crystal

§8-a. Final properties of B(∞). Now we follow the ideas of [G1, §13].

Lemma 8.1. Let M ∈ Rep_{I} H_{m} be irreducible.

(i) For any i ∈ I, either ε_{i}(f̃_{i}^{∗}M) = ε_{i}(M) or ε_{i}(M) + 1.

(ii) For any i, j ∈ I with i ≠ j, ε_{i}(f̃_{j}^{∗}M) = ε_{i}(M). Proof. We prove (i), the proof of (ii) being similar. By the Shuffle Lemma, we certainly have that ε_{i}(f̃_{i}^{∗}M) ≤ ε_{i}(M) + 1. Now let N = f̃_{i}^{∗}M. Then obviously, ε_{i}(ẽ_{i}^{∗}N) ≤ ε_{i}(N). Hence, ε_{i}(M) ≤ ε_{i}(f̃_{i}^{∗}M).

Lemma 8.2. Let M ∈ Rep_{I} H_{m} be irreducible and i, j ∈ I. If ε_{i}(f̃_{j}^{∗}M) = ε_{i}(M) then, writing ε := ε_{i}(M), we have ẽ_{i}^{ε}f̃_{j}^{∗}M ≅ f̃_{j}^{∗}ẽ_{i}^{ε}M. Proof. Set n = m − ε. Let N = ẽ_{i}^{ε}M, so N is an irreducible H_{n}-module with ε_{i}(N) = 0 and M = f̃_{i}^{ε}N. For 0 ≤ b ≤ ε, let Q_{b} = res_{i}^{ε−b}f̃_{j}^{∗}M. Theorem 6.11(i) and Lemma 6.3 imply that in the Grothendieck group, Q_{b} is some number of copies of ẽ_{i}^{ε−b}f̃_{j}^{∗}M plus terms with strictly smaller ε_{i}. In particular, ε_{i}(L) ≤ b for all composition factors L of Q_{b}, while Q_{0} is consists only of copies of ẽ_{i}^{ε}f̃_{j}^{∗}M.

We will show by decreasing induction on b = ε, ε − 1, …, 0 that there is a non-zero H_{n+b+1}-module homorphism

γ_{b} : ind_{1,n,b}^{n+b+1}L(j) ⊠ N ⊠ L(i^{b}) → Q_{b}.

In case b = ε, Q_{ε} = f̃_{j}^{∗}M is a quotient of ind_{1,m}^{m+1}L(j) ⊠ M and M is a quotient of ind_{n,ε}^{m}N ⊠ L(i^{ε}), so the induction starts. Now we suppose by induction that we have proved γ_{b} ≠ 0 exists for b ≥ 1 and construct γ_{b−1}.

Consider res_{n+b,1}^{n+b+1}ind_{1,n,b}^{n+b+1}L(j)⊠N ⊠L(i^{b}). By the Mackey Theorem, this has a filtration 0 ⊂ F_{1} ⊂ F_{2} ⊂ F_{3} with successive quotients

F_{1} ≃ ind_{1,n,b−1,1}^{n+b,1}res_{1,n,b−1,1}^{1,n,b}L(j) ⊠ N ⊠ L(i^{b}),
F_{2}/F_{1} ≃ ind_{1,n−1,b,1}^{n+b,1}^{w}res_{1,n−1,1,b}^{1,n,b}L(j) ⊠ N ⊠ L(i^{b}),
F_{3}/F_{2} ≃ ind_{n,b,1}^{n+b,1}N ⊠ L(i^{b}) ⊠ L(j),

where w is the obvious permutation. As γ_{b} ≠ 0, Frobenius reciprocity implies that there is a copy of the H_{1,n,b}-module L(j) ⊛ N ⊛ L(i^{b}) in the image of γ_{b}. Now b > 0, so the q(i)-eigenspace of X_{n+b+1} + X_{n+b+1}^{−1} acting on L(j) ⊛ N ⊛ L(i^{b}) is non-trivial. We conclude that the map

γ̃_{b} = res_{i}(γ_{b}) : res_{i}ind_{1,n,b}^{n+b+1}L(j) ⊠ N ⊠ L(i^{b}) → res_{i}Q_{b} = Q_{b−1}

is non-zero.

If i ≠ j, then it follows from the description of F_{3}/F_{2} and F_{2}/F_{1} above that res_{i}(F_{3}/F_{1}) = 0. So in this case, we necessarily have that γ̃_{b}(F_{1}) ≠ 0. Similarly if i = j, res_{i}(F_{2}/F_{1}) = 0, so if γ̃_{b}(F_{1}) = 0 we see that γ̃_{b} factors to a non-zero homomorphism

res_{n+b}^{n+b,1}ind_{n,b,1}^{n+b,1}N ⊠ L(i^{b}) ⊠ L(i) → Q_{b−1}.

But this implies that Q_{b−1} has a constituent L with ε_{i}(L) = b, which we know is not the case. Hence we have that γ̃_{b}(F_{1}) ≠ 0 in the case i = j too.

Hence, the restriction of γ̃_{b} to F_{1} gives us a non-zero homomorphism

res_{n+b}^{n+b,1}ind_{1,n,b−1,1}^{b+n,1}res_{1,n,b−1,1}^{1,n,b}L(j) ⊠ N ⊠ L(i^{b}) → Q_{b−1}.

Now finally as all composition factors of res_{b−1}^{b}L(i^{b}) are isomorphic to L(i^{b−1}), this implies the existence of a non-zero homomorphism γ_{b−1} : ind_{1,n,b−1}^{b+n}L(j) ⊠ N ⊠ L(i^{b−1}) → Q_{b−1} completing the induction.

Now taking b = 0 we have a non-zero map γ_{0} : ind_{1,n}^{n+1}L(j) ⊠ N → Q_{0}. But the left hand side has irreducible cosocle f̃_{j}^{∗}N = f̃_{j}^{∗}ẽ_{i}^{ε}M while all composition factors of the right hand side are isomorphic to ẽ_{i}^{ε}f̃_{j}^{∗}M. This completes the proof.

Corollary 8.3. Let M ∈ Rep_{I}(H_{m}) be irreducible and i ∈ I. Let M_{1} = ẽ_{i}^{ε_{i}(M)}M and M_{2} = (ẽ_{i}^{∗})^{ε_{i}^{∗}(M)}M. Then, ε_{i}^{∗}(M) = ε_{i}^{∗}(M_{1}) if and only if ε_{i}(M) = ε_{i}(M_{2}). Proof. Since we can apply the automorphism σ, it suffices to check just one of the implications. So suppose ε_{i}(M) = ε_{i}(M_{2}). Clearly, ε_{i}^{∗}(M) ≥ ε_{i}^{∗}(M_{1}). For the reverse inequality, the preceeding two lemmas show that

M_{1} ≅ ẽ_{i}^{ε_{i}(M)}M ≅ ẽ_{i}^{ε_{i}(M)}(f̃_{i}^{∗})^{ε_{i}^{∗}(M)}M_{2} ≅ (f̃_{i}^{∗})^{ε_{i}^{∗}(M)}ẽ_{i}^{ε_{i}(M)}M_{2}.

This shows that ε_{i}^{∗}(M_{1}) ≥ ε_{i}^{∗}(M).

Lemma 8.4. Let M ∈ Rep_{I} H_{m} be irreducible and i ∈ I satisfy ε_{i}(f̃_{i}^{∗}M) = ε_{i}(M) + 1. Then ẽ_{i}f̃_{i}^{∗}M = M. Proof. Set ε := ε_{i}(M) and N = ẽ_{i}^{ε}M. By the Shuffle Lemma and Theorem 5.12,

[ind_{1,m−ε,ε}^{m+1}L(i) ⊛ N ⊛ L(i^{ε})] = [ind_{m−ε,ε+1}^{m+1}N ⊛ L(i^{ε+1})].

Hence by Theorem 6.11(i), [ind_{1,m−ε,ε}^{m+1}L(i) ⊛ N ⊛ L(i^{ε})] equals [f̃_{i}^{ε+1}N] = [f̃_{i}M] plus terms [L] for irreducible L with ε_{i}(L) ≤ ε. On the other hand, ind_{1,m−ε,ε}^{m+1}L(i) ⊛ N ⊛ L(i^{ε}) surjects onto f̃_{i}^{∗}M. So the assumption ε_{i}(f̃_{i}^{∗}M) = ε + 1 implies f̃_{i}M ≅ f̃_{i}^{∗}M.

§8-b. Crystals. Let us now recall some definitions from [Ka]. A crystal is a set B endowed with maps

φ_{i}, ε_{i} : B → ℤ ∪ {−∞} (i ∈ I),
ẽ_{i}, f̃_{i} : B → B ∪ {0} (i ∈ I),
wt : B → P

such that (C1) φ_{i}(b) = ε_{i}(b) + ⟨h_{i}, wt(b)⟩ for any i ∈ I; (C2) if b ∈ B satisfies ẽ_{i}b ≠ 0, then ε_{i}(ẽ_{i}b) = ε_{i}(b) − 1, φ_{i}(ẽ_{i}b) = φ_{i}(b) + 1 and wt(ẽ_{i}b) =

wt(b) + α_{i}; (C3) if b ∈ B satisfies f̃_{i}b ≠ 0, then ε_{i}(f̃_{i}b) = ε_{i}(b) + 1, φ_{i}(f̃_{i}b) = φ_{i}(b) − 1 and wt(ẽ_{i}b) =

wt(b) − α_{i}; (C4) for b_{1}, b_{2} ∈ B, b_{2} = f̃_{i}b_{1} if and only if b_{1} = ẽ_{i}b_{2};

(C5) if φ_{i}(b) = −∞, then ẽ_{i}b = f̃_{i}b = 0. For example, for each i ∈ I, we have the crystal B_{i} defined as a set to be {b_{i}(n) | n ∈ ℤ} with

ε_{j}(b_{i}(n)) = \{ \begin{matrix}−n & if j = i, \\ −∞ & if j ≠ i; \end{matrix}  φ_{j}(b_{i}(n)) = \{ \begin{matrix}n & if j = i, \\ −∞ & if j ≠ i; \end{matrix}
ẽ_{j}(b_{i}(n)) = \{ \begin{matrix}b_{i}(n + 1) & if j = i, \\ 0 & if j ≠ i; \end{matrix}  f̃_{j}(b_{i}(n)) = \{ \begin{matrix}b_{i}(n − 1) & if j = i, \\ 0 & if j ≠ i; \end{matrix}

and wt(b_{i}(n)) = nα_{i}. We abbreviate b_{i}(0) by b_{i}. Also for λ ∈ P, we have the crystal T_{λ} equal as a set to {t_{λ}}, with ε_{i}(t_{λ}) = φ_{i}(t_{λ}) = −∞, ẽ_{i}t_{λ} = f̃_{i}t_{λ} = 0 and wt(t_{λ}) = λ.

A morphism ψ : B → B′ of crystals is a map ψ : B ∪ {0} → B′ ∪ {0} such that (H1) ψ(0) = 0; (H2) if ψ(b) ≠ 0 for b ∈ B, then wt(ψ(b)) = wt(b), ε_{i}(ψ(b)) = ε_{i}(b) and φ_{i}(ψ(b)) = ψ(b); (H3) for b ∈ B such that ψ(b) ≠ 0 and ψ(ẽ_{i}b) ≠ 0, we have that ψ(ẽ_{i}b) = ẽ_{i}ψ(b); (H4) for b ∈ B such that ψ(b) ≠ 0 and ψ(f̃_{i}b) ≠ 0, we have that ψ(f̃_{i}b) = f̃_{i}ψ(b).

A morphism of crystals is called strict if ψ commutes with the ẽ_{i}’s and f̃_{i}’s, and an embedding if ψ is injective.

We also need the notion of a tensor product of two crystals B, B′. As a set, B ⊗ B′ is equal to {b ⊗ b′ | b ∈ B, b′ ∈ B′}. This is made into a crystal by

ε_{i}(b ⊗ b′) = max(ε_{i}(b), ε_{i}(b′) − ⟨h_{i}, wt(b)⟩), φ_{i}(b ⊗ b′) = max(φ_{i}(b) + ⟨h_{i}, wt(b′)⟩, φ_{i}(b′)),
ẽ_{i}(b ⊗ b′) = \{ \begin{matrix}ẽ_{i}b ⊗ b′ & if φ_{i}(b) ≥ ε_{i}(b′) \\ b ⊗ ẽ_{i}b′ & if φ_{i}(b) < ε_{i}(b′) \end{matrix} , f̃_{i}(b ⊗ b′) = \{ \begin{matrix}f̃_{i}b ⊗ b′ & if φ_{i}(b) > ε_{i}(b′) \\ b ⊗ f̃_{i}b′ & if φ_{i}(b) ≤ ε_{i}(b′) \end{matrix} ,
wt(b ⊗ b′) = wt(b) + wt(b′).

Here, we understand b ⊗ 0 = 0 = 0 ⊗ b.

Having recalled these definitions, we now explain how to make our sets B(∞) and B(λ) from (5.5) and (5.12) into crystals in the above sense. To do this, it just remains to define the weight functions on both B(∞) and B(λ), as well as the function φ_{i} on B(∞): set

wt(M) = −γ,  (8.1)

for an irreducible M ∈ Rep_{γ} H_{n}, and

wt^{λ}(N) = λ − γ,  (8.2)

for an irreducible N ∈ Rep_{γ} H_{n}^{λ}. Also for an irreducible [M] ∈ B(∞), define

φ_{i}(M) = ε_{i}(M) + ⟨h_{i}, wt(M)⟩,  (8.3)

so that property (C1) is automatic. Thus we have defined (B(∞), ε_{i}, φ_{i}, ẽ_{i}, f̃_{i}, wt) and (B(λ), ε_{i}, φ_{i}, ẽ_{i}^{λ}, f̃_{i}^{λ}, wt^{λ}) purely in terms of the representation theory of the Hecke-Clifford superalgebras.

Lemma 8.5. (B(∞), ε_{i}, φ_{i}, ẽ_{i}, f̃_{i}, wt) and each (B(λ), ε_{i}, φ_{i}, ẽ_{i}^{λ}, f̃_{i}^{λ}, wt^{λ}) for λ ∈ P_{+} are crystals in the sense of Kashiwara. Proof. Property (C1) is Lemma 6.25 or the definition in the affine case. Property (C4) is Lemma 5.10. The remaining properties are immediate.

Recall the embedding infl^{λ} : B(λ) ∪ {0} → B(∞) ∪ {0} from (5.13).

Lemma 8.6. The map B(λ) ↪ B(∞) ⊗ T_{λ}, [N] ↦ infl^{λ}[N] ⊗ t_{λ} is an embedding of crystals with image

{[M] ⊗ t_{λ} ∈ B(∞) ⊗ T_{λ} | ε_{i}^{∗}(M) ≤ ⟨h_{i}, λ⟩ for each i ∈ I} .

Proof. Since ẽ_{i}^{λ} and f̃_{i}^{λ} are restrictions of ẽ_{i}, f̃_{i} from B(∞) to B(λ), respectively, the first statement is immediate. The second is a restatement of Corollary 6.13.

§8-c. Identification of B(∞) and B(λ). The first lemma follows directly from Lemmas 8.1 and 8.2, applying the automorphism σ to get (ii).

Lemma 8.7. Let M ∈ Rep_{I} H_{m} be irreducible and i, j ∈ I with i ≠ j. Set a = ε_{i}^{∗}(M). (i) ε_{j}(M) = ε_{j}((ẽ_{i}^{∗})^{a}M). (ii) If ε_{j}(M) > 0, then ε_{i}^{∗}(ẽ_{j}M) = ε_{i}^{∗}(M) and (ẽ_{i}^{∗})^{a}ẽ_{j}M ≅ ẽ_{j}(ẽ_{i}^{∗})^{a}M.

The proof of the next result is taken from [G1, Proposition 10.2].

Lemma 8.8. Let M ∈ Rep_{I} H_{m} be irreducible and i ∈ I. Set a = ε_{i}^{∗}(M) and M̄ = (ẽ_{i}^{∗})^{a}M. (i) ε_{i}(M) = max(ε_{i}(M̄), a − ⟨h_{i}, wt(M̄)⟩). (ii) If ε_{i}(M) > 0,

ε_{i}^{∗}(ẽ_{i}M) = \{ \begin{matrix}a & if ε_{i}(M̄) ≥ a − ⟨h_{i}, wt(M̄)⟩, \\ a − 1 & if ε_{i}(M̄) < a − ⟨h_{i}, wt(M̄)⟩. \end{matrix}

(iii) If ε_{i}(M) > 0,

(ẽ_{i}^{∗})^{b}ẽ_{i}M ≅ \{ \begin{matrix}ẽ_{i}(M̄) & if ε_{i}(M̄) ≥ a − ⟨h_{i}, wt(M̄)⟩, \\ M̄ & if ε_{i}(M̄) < a − ⟨h_{i}, wt(M̄)⟩, \end{matrix}

where b = ε_{i}^{∗}(ẽ_{i}M).

Proof. Let ε = ε_{i}(M), n = m − ε, and N = (ẽ_{i})^{ε}M.

(i) By twisting with σ, it suffices to prove that (for arbitrary M)

ε_{i}^{∗}(M) = max(ε_{i}^{∗}(N), ε − ⟨h_{i}, wt(N)⟩).  (8.4)

Define the weights λ(0), λ(1), ⋯ ∈ P_{+} by taking the Λ_{i}-coefficient of λ(r) to be ε_{i}^{∗}(N) + r and the Λ_{j}-coefficients of λ(r) for j ≠ i to be ≫ 0. Then I_{λ(r)}N = 0 for any r ≥ 0, see Corollory 6.13. Moreover, the same corollary implies that for k = φ_{i}^{λ(r)}(N),

ε_{i}^{∗}(f̃_{i}^{k}N) = ⟨h_{i}, λ(r)⟩ and ε_{i}^{∗}(f̃_{i}^{k+1}N) = ⟨h_{i}, λ(r)⟩ + 1.

Now, φ_{i}^{λ}(N) − ε_{i}^{λ}(N) = ⟨h_{i}, λ + wt(N)⟩ and ε_{i}(N) = ε_{i}^{λ}(N). Moreover, by Lemma 8.1(i) (twisted with σ) we have ε_{i}^{∗}(f̃^{k}N) ≥ ε_{i}^{∗}(N) for any k. All of these applied consecutively to λ(0), λ(1), … imply:

ε_{i}^{∗}(f̃_{i}^{s}N) = \{ \begin{matrix}ε_{i}^{∗}(N) & if s ≤ ε_{i}^{∗}(N) + ε_{i}(N) + ⟨h_{i}, wt(N)⟩, \\ s − ε_{i}(N) − ⟨h_{i}, wt(N)⟩ & if s ≥ ε_{i}^{∗}(N) + ε_{i}(N) + ⟨h_{i}, wt(N)⟩ \end{matrix}

for all s ≥ 0. For s = ε, taking into account ε_{i}(N) = 0, this gives (8.4).

(ii) Observe by (8.4) that ε_{i}^{∗}(ẽ_{i}M) = ε_{i}^{∗}(M)−1 if and only if ε > ⟨h_{i}, wt(N)⟩+ε_{i}^{∗}(N), and that otherwise ε_{i}^{∗}(ẽ_{i}M) = ε_{i}^{∗}(M). But ⟨h_{i}, wt(N)⟩ = ⟨h_{i}, wt(M)⟩ + 2ε and ⟨h_{i}, wt(M̄)⟩ = ⟨h_{i}, wt(M)⟩ + 2a so (ii) follows if we show that ⟨h_{i}, wt(M)⟩ + ε_{i}^{∗}(N) + ε < 0 if and only if ⟨h_{i}, wt(M)⟩ + ε_{i}(M̄) + a < 0. But by (i) and (8.4),

⟨h_{i}, wt(M)⟩ + ε_{i}^{∗}(N) + ε = max(⟨h_{i}, wt(M)⟩ + ε_{i}^{∗}(N) + ε_{i}(M̄), ε_{i}^{∗}(N) − a),
⟨h_{i}, wt(M)⟩ + ε_{i}(M̄) + a = max(⟨h_{i}, wt(M)⟩ + ε_{i}(M̄) + ε_{i}^{∗}(N), ε_{i}(M̄) − ε).

Moreover, obviously ε_{i}^{∗}(N) − a ≤ 0 and ε_{i}(M̄) − ε ≤ 0, and it remains to observe that ε_{i}^{∗}(N) − a = 0 if and only if ε_{i}(M̄) − ε = 0, thanks to Corollary 8.3.

(iii) This follows from (ii) and Lemmas 8.2 and 8.4 (twisted with σ).

Now for each i ∈ I, define a map

Ψ_{i} : B(∞) → B(∞) ⊗ B_{i}  (8.5)

mapping each [M] ∈ B(∞) to [(ẽ_{i}^{∗})^{a}M] ⊗ f̃_{i}^{a}b_{i}, where a = ε_{i}^{∗}(M).

Lemma 8.9. The following properties hold:

(i) for every [M] ∈ B(∞), wt(M) is a negative sum of simple roots; (ii) [1] is the unique element of B(∞) with weight 0; (iii) ε_{i}(1) = 0 for every i ∈ I; (iv) ε_{i}(M) ∈ ℤ for every [M] ∈ B(∞) and every i ∈ I;

(v) for every i, the map Ψ_{i} : B(∞) → B(∞) ⊗ B_{i} defined above is a strict embedding of crystals;

(vi) Ψ_{i}(B(∞)) ⊆ B(∞) × {f̃_{i}^{n}b_{i} | n ≥ 0};

(vii) for any [M] ∈ B(∞) other than [1], there exists i ∈ I such that Ψ_{i}([M]) = [N]⊗f̃_{i}^{n}b_{i}

for some [N] ∈ B(∞) and n > 0. Proof. Properties (i)–(iv) are immediate from our construction of B(∞). The information required to verify (v) is exactly contained in Lemmas 8.7 and 8.8. Finally, (vi) is immediate from the definition of Ψ_{i}, and (vii) holds because every such M has ε_{i}^{∗}(M) > 0 for at least one i ∈ I.

The properties in Lemma 8.9 exactly characterize the crystal B(∞) by [KS, Proposition 3.2.3]. Hence, we have proved:

Theorem 8.10. The crystal B(∞) is isomorphic to Kashiwara’s crystal B(∞) associated to the crystal base of U_{ℚ}^{−}.

In view of [Ka, Theorem 8.2], we can also identify our maps Ψ_{i} with those of [Ka]. Taking into account [Ka, Proposition 8.1] we can then identify our functions ε_{i}^{∗} on B(∞) with those in [Ka]. It follows from this, Lemma 8.6 and [Ka, Proposition 8.2] that:

Theorem 8.11. For each λ ∈ P_{+}, the crystal B(λ) is isomorphic to Kashiwara’s crystal B(λ) associated to the integrable highest weight U_{ℚ}-module of highest weight λ.

§8-d. Blocks. Let λ ∈ P_{+}. As an application of the theory, we mention here another delightful argument of Grojnowski from [G2], which in our setting classifies the blocks of the cyclotomic Hecke-Clifford superalgebras H_{n}^{λ}.

Let us recall the definition in the case of a finite dimensional superalgebra A. Let ∼ be the equivalence relation on the set of isomorphism classes of irreducible A-modules such that [L] ∼ [M] if and only if there exists a chain L ≅ L_{0}, L_{1}, …, L_{m} ≅ M of irreducible A-modules with either Ext_{A}^{1}(L_{i}, L_{i+1}) ≠ 0 or Ext_{A}^{1}(L_{i+1}, L_{i}) ≠ 0 for each i. Given a ∼- equivalence class b, the corresponding block Rep_{b} A is the full subcategory of Rep A consisting of the A-modules all of whose composition factors belong to the equivalence class ∼. Thus, there is a decomposition

Rep A = ⨁_{b} Rep_{b} A

as b runs over all ∼-equivalence classes. As usual, there are various equivalent points of view: for instance, [L] ∼ [M] if and only if the even central characters χ_{L}, χ_{M} : Z(A)¯0 → F arising from the actions on the irreducible modules L, M are equal. Alternatively, the blocks of A can be defined in terms of a decomposition of the identity 1_{A} into a sum of centrally primitive even idempotents.

Theorem 8.12. Let M, N be irreducible H_{n}^{λ}-modules with M ≅ N̸. Let

0 ⟶ M ⟶ X ⟶ N ⟶ 0

be an exact sequence of H_{n}-modules. Then, pr^{λ}X = X and the sequence is also an exact sequence of H_{n}^{λ}-modules. Proof. We note that for irreducible modules M, N in Rep_{I} H_{n} with M ≅ N̸, we have that

Hom_{H_{n−1}}(res_{n−1}^{n}M, res_{n−1}^{n}N) = 0.

This follows immediately from Corollary 6.12, Lemma 6.3 and (6.6). Using this in place of [G2, Lemma 2], the proof of the theorem is now completed by exactly the same argument as in [G2].

Note the theorem can be reinterpreted as saying that

Ext_{H_{n}}^{1}(M, N) ≃ Ext_{H_{n}^{λ}}^{1}(M, N)  (8.6)

for irreducible H_{n}^{λ}-modules M, N with M ≅ N̸. This is certainly not the case if M ≅ N! Recalling the definitions from §4-f, we immediately deduce the following corollary which determines the blocks of H_{n}^{λ}:

Corollary 8.13. The blocks of H_{n}^{λ} are precisely the subcategories Rep_{γ} H_{n}^{λ} for γ ∈ Γ_{n}. Moreover, Rep_{γ} H_{n}^{λ} is non-trivial if and only if the (λ−γ)-weight space of the highest weight module K(λ)_{ℚ} is non-zero.

9. Branching rules

§9-a. Kang’s description of the crystal graph. Now we focus on the fundamental highest weight Λ_{0}. In this case, Kang [Kan] has given a convenient combinatorial description of the crystal B(Λ_{0}) in terms of Young diagrams, which we now describe.

For any n ≥ 0, let λ = (λ_{1}, λ_{2}, …) be a partition of n, i.e. a non-increasing sequence of non-negative integers summing to n. Recall that ℓ ∈ ℤ_{>0} ∪ {∞} and h = 2ℓ + 1. We call λ an h-strict partition if h divides λ_{r} whenever λ_{r} = λ_{r+1} for r ≥ 1 (to interpret correctly in case h = ∞, we adopt the convention that ∞ divides 0 and nothing else). Let 𝒫_{h}(n) denote the set of all h-strict partitions of n, and 𝒫_{h} := ⋃_{n≥0} 𝒫_{h}(n). We say that λ ∈ 𝒫_{h}(n) is restricted if in addition

\{ \begin{matrix}λ_{r} − λ_{r+1} < h & if h|λ_{r}, \\ λ_{r} − λ_{r+1} ≤ h & if h ∤ λ_{r} \end{matrix}

for each r ≥ 1. Let ℛ𝒫 _{h}(n) denote the set of all restricted h-strict partitions of n, and ℛ𝒫 _{h} := ⋃_{n≥0} ℛ𝒫 _{h}(n).

Let λ ∈ 𝒫_{h} be an h-strict partition. We identify λ with its Young diagram

λ = {(r, s) ∈ ℤ_{>0} × ℤ_{>0} | s ≤ λ_{r}}.

Elements (r, s) ∈ ℤ_{>0} × ℤ_{>0} are called nodes. We label the nodes of λ with residues, which are the elements of the set I = {0, 1, …, ℓ}. The labelling depends only on the column and follows the repeating pattern

0, 1, …, ℓ − 1, ℓ, ℓ − 1, …, 1, 0,

starting fom the first column and going to the right, see Example 9.1 below. The residue of the node A is denoted res A. Define the residue content of λ to be the tuple

cont(λ) = (γ_{i})_{i∈I}  (9.1)

where for each i ∈ I, γ_{i} is the number of nodes of residue i contained in the diagram λ.

Let i ∈ I be some fixed residue. A node A = (r, s) ∈ λ is called i-removable (for λ) if one of the following holds:

(R1) res A = i and λ_{A} := λ − {A} is again an h-strict partition;

(R2) the node B = (r, s+1) immediately to the right of A belongs to λ, res A = res B = i,

and both λ_{B} = λ − {B} and λ_{A,B} := λ − {A, B} are h-strict partitions. Similarly, a node B = (r, s) ∉ λ is called i-addable (for λ) if one of the following holds:

(A1) res B = i and λ^{B} := λ ∪ {B} is again an h-strict partition;

(A2) the node A = (r, s − 1) immediately to the left of B does not belong to λ, res A =

res B = i, and both λ^{A} = λ ∪ {A} and λ^{A,B} := λ ∪ {A, B} are h-strict partitions. We note that (R2) and (A2) above are only possible in case i = 0.

Now label all i-addable nodes of the diagram λ by + and all i-removable nodes by −. Then, the i-signature of λ is the sequence of pluses and minuses obtained by going along the rim of the Young diagram from bottom left to top right and reading off all the signs. The reduced i-signature of λ is obtained from the i-signature by successively erasing all neighbouring pairs of the form +−.

Note the reduced i-signature always looks like a sequence of −’s followed by +’s. Nodes corresponding to a − in the reduced i-signature are called i-normal, nodes corresponding to a + are called i-conormal. The rightmost i-normal node (corresponding to the rightmost − in the reduced i-signature) is called i-good, and the leftmost i-conormal node (corresponding to the leftmost + in the reduced i-signature) is called i-cogood.

Example 9.1. Let h = 5, so ℓ = 2. The partition λ = (16, 11, 10, 10, 9, 5, 1) belongs to ℛ𝒫 _{h}, and its residues are as follows:

0 1 2 1 0 0 1 2 1 0 0 1 2 1 0 0
0 1 2 1 0 0 1 2 1 0 0
0 1 2 1 0 0 1 2 1 0
0 1 2 1 0 0 1 2 1 0
0 1 2 1 0 0 1 2 1
0 1 2 1 0

0

The 0-addable and 0-removable nodes are as labelled in the diagram:

[figure: − − ; − ; − ; −; +; +]

Hence, the 0-signature of λ is −, −, +, +, −, −, − and the reduced 0-signature is −, −, −. Note the nodes corresponding to the −’s in the reduced 0-signature have been circled in the above diagram. So, there are three 0-normal nodes, the rightmost of which is 0-good; there are no 0-conormal or 0-cogood nodes.

In general, we define

ε_{i}(λ) = ♯{i-normal nodes in λ} = ♯{−’s in the reduced i-signature of λ},  (9.2)
φ_{i}(λ) = ♯{i-conormal nodes in λ} = ♯{+’s in the reduced i-signature of λ}.  (9.3)

Also set

ẽ_{i}(λ) = \{ \begin{matrix}λ_{A} & if ε_{i}(λ) > 0 and A is the (unique) i-good node, \\ 0 & if ε_{i}(λ) = 0, \end{matrix}  (9.4)
f̃_{i}(λ) = \{ \begin{matrix}λ^{B} & if φ_{i}(λ) > 0 and B is the (unique) i-cogood node, \\ 0 & if φ_{i}(λ) = 0. \end{matrix}  (9.5)

Finally define

wt(λ) = Λ_{0} − ∑_{i∈I} γ_{i}α_{i}  (9.6)

where cont(λ) = (γ_{i})_{i∈I}. We have now defined a datum (𝒫_{h}, ε_{i}, φ_{i}, ẽ_{i}, f̃_{i}, wt) which makes the set 𝒫_{h} of all h-strict partitions into a crystal in the sense of §8-b. The definitions imply that ẽ_{i}(λ), f̃_{i}(λ) are restricted (or zero) in case λ is itself restricted. Hence, ℛ𝒫 _{h} is a sub-crystal of 𝒫_{h}. We can now state the main result of Kang [Kan, 7.1] for type A_{2ℓ}^{(2)}:

Theorem 9.2. The set ℛ𝒫 _{h} equipped with ε_{i}, φ_{i}, ẽ_{i}, f̃_{i}, wt as above is isomorphic (in the unique way) to the crystal B(Λ_{0}) associated to the integrable highest weight U_{ℚ}-module of fundamental highest weight Λ_{0}.

Example 9.3. The crystal graph of ℛ𝒫 _{h} = B(Λ_{0}) in case h = 3, up to degree 10, is as follows:

[figure: ∅; 0; 0 1; 0 1 0; 0 1 0 0; 0 1 0 0 0; 0 1 0 0 1; 0 1 0 0 0 1; 0 1 0 0 1 0; 0 1 0 0 0 1 0; 0 1 0 0 1 0 0; 0 1 0 0 1 0 1; 0 1 0 0 0 1 0 0; 0 1 0 0 1 0 1 0; 0 1 0 0 1 0 0 1; 0 1 0 0 0 1 0 0 1; 0 1 0 0 1 0 1 0 0; 0 1 0 0 1 0 0 1 0; 0 1 0 0 1 0 1 0 0 0; 0 1 0 0 1 0 1 0 0 1; 0 1 0 0 0 1 0 0 1 0; 0 1 0 0 1 0 0 1 0 0; 0; 1; 0; 0; 0; 1; 0; 1; 0; 0; 1; 1; 0; 0; 0; 1; 1; 0; 0; 0; 1; 0]

⋮

Let us also mention here the extension of Morris’ notion of h-bar core [Mo] to an arbitrary h-strict partition λ ∈ 𝒫_{h}(n). By an h-bar of λ, we mean one of the following:

(B1) the rightmost h nodes of row i of λ if λ_{i} ≥ h and either h|λ_{i} or λ has no row of

length (λ_{i} − h); (B2) the set of nodes in rows i and j of λ if λ_{i} + λ_{j} = h.

If λ has no h-bars, it is called an h-bar core. In general, the h-bar core λ̃ of λ is obtained by successively removing h-bars, reordering the rows each time so that the result still lies in 𝒫_{h}, until it is reduced to a core. The h-bar weight of λ, denoted w(λ), is then the total number of h-bars that get removed. For a Lie theoretic explanation of these notions, we refer the reader to [Kc, §12.6]. In particular, as observed in [LT, §4], for μ, λ ∈ 𝒫_{h}(n) we have that

cont(μ) = cont(λ) if and only if μ̃ = λ̃.  (9.7)

Also, bearing in mind Theorem 9.2, we can state Kac’ formula [Kc, (12.13.5)] for the character of the highest weight U_{ℚ}-module of highest weight Λ_{0} as follows: for λ ∈ ℛ𝒫 _{h}(n),

♯{μ ∈ ℛ𝒫 _{h}(n) | cont(μ) = cont(λ)} = Par_{ℓ}(w(λ)),  (9.8)

where Par_{ℓ}(N) denotes the number of partitions of N as a sum of positive integers of ℓ different colors.

§9-b. Representations of finite Hecke-Clifford superalgebras. Now that we have an explicit description of the crystal B(Λ_{0}), we formulate a more combinatorial description of our main results for the representation theory of the finite Hecke-Clifford superalgebras H_{n}^{fin}. Recall from Remark 4.2 that this is precisely the cyclotomic Hecke-Clifford superalgebra H_{n}^{Λ_{0}}. The results of this subsection also hold in the degenerate case, when H_{n}^{fin} is the finite Sergeev superalgebra, see §2-k.

As explained in §5-d, the isomorphism classes of irreducible H_{n}^{fin}-modules are parametrized by the nodes of the crystal graph B(Λ_{0}). By Theorems 9.2 and 8.11, we can identify B(Λ_{0}) with ℛ𝒫 _{h}. In other words, we can use the set ℛ𝒫 _{h}(n) of restricted h-strict partitions of n to parametrize the irreducible H_{n}^{fin}-modules for each n ≥ 0. Let us write M(λ) for the irreducible H_{n}^{fin}-module corresponding to λ ∈ ℛ𝒫 _{h}(n). To be precise,

M(λ) := L(i_{1}, …, i_{n})

if λ = f̃_{i_{n}} … f̃_{i_{1}}∅. Here the operator f̃_{i} is as defined in (9.5), corresponding under the identification ℛ𝒫 _{h}(n) = B(Λ_{0}) to the crystal operator denoted f̃_{i}^{Λ_{0}} in earlier sections, and ∅ denotes the empty partition, corresponding to 1_{λ} ∈ B(Λ_{0}).

For λ ∈ ℛ𝒫 _{h}(n), we also define

b(λ) := ♯{r ≥ 1 | h ∤ λ_{r}},  (9.9)

the number of parts of λ that are not divisible by h. The definition of residue content immediately gives that

b(λ) ≡ γ_{0} (mod 2),  (9.10)

where γ_{0} denotes the number of 0’s in the residue content of λ.

Theorem 9.4. The modules {M(λ) | λ ∈ ℛ𝒫 _{h}(n)} form a complete set of pairwise nonisomorphic irreducible H_{n}^{fin}-modules. Moreover, for λ, μ ∈ ℛ𝒫 _{h}(n),

(i) M(λ) ≅ M(λ)^{τ}; (ii) M(λ) is of type M if b(λ) is even, type Q if b(λ) is odd; (iii) M(μ) and M(λ) belong to the same block if and only if cont(μ) = cont(λ); (iv) M(λ) is a projective module if and only if λ is an h-bar core.

Proof. We have already discussed the first statement of the theorem, being a consequence of our main results combined with Kang’s Theorem 9.2. For the rest, (i) follows from Corollary 5.13, (ii) is a special case of Lemma 5.14 combined with (9.10), and (iii) is a special case of Corollary 8.13. For (iv), note that if M(λ) is projective then it is the only irreducible in its block, hence by (9.8), Par_{ℓ}(w(λ)) = 1. So either w(λ) = 0, or ℓ = 1 and w(λ) = 1. Now if w(λ) = 0 then λ is an h-bar core so the Shapovalov form on the (1-dimensional) wt(λ)-weight space of K(Λ_{0})_{ℤ} is 1 (since wt(λ) is conjugate to Λ_{0} under the action of the affine Weyl group). Hence, M(λ) is projective by Theorem 7.16(ii). To rule out the remaining possibilty ℓ = 1 and w(λ) = 1, one checks in that case that the Shapovalov form on the wt(λ)-weight space of K(Λ_{0})_{ℤ} is 3.

The next two theorems summarize earlier results concerning restriction and induction.

Theorem 9.5. Let λ ∈ ℛ𝒫 _{h}(n). There exist H_{n−1}^{fin}-modules e_{i}M(λ) for each i ∈ I, unique up to isomorphism, such that

(i) res_{H_{n−1}^{fin}}^{H_{n}^{fin}}M(λ) ≅ \{ \begin{matrix}2e_{0}M(λ) ⊕ 2e_{1}M(λ) ⊕ ⋯ ⊕ 2e_{ℓ}M(λ) & if b(λ) is odd, \\ e_{0}M(λ) ⊕ 2e_{1}M(λ) ⊕ ⋯ ⊕ 2e_{ℓ}M(λ) & if b(λ) is even; \end{matrix}

(ii) for each i ∈ I, e_{i}M(λ) ≠ 0 if and only if λ has an i-good node A, in which case e_{i}M(λ) is a self-dual indecomposable module with irreducible socle and cosocle isomorphic to M(λ_{A}).

Moreover, if i ∈ I and λ has an i-good node A, then

(iii) the multiplicity of M(λ_{A}) in e_{i}M(λ) is ε_{i}(λ), ε_{i}(λ_{A}) = ε_{i}(λ)−1, and ε_{i}(μ) < ε_{i}(λ)−1

for all other composition factors M(μ) of e_{i}M(λ); (iv) End_{H_{n−1}^{fin}}(e_{i}M(λ)) ≃ End_{H_{n−1}^{fin}}(M(λ_{A}))^{⊕ε_{i}(λ)} as a vector superspace;

(v) Hom_{H_{n−1}^{fin}}(e_{i}M(λ), e_{i}M(μ)) = 0 for all μ ∈ ℛ𝒫 _{h}(n) with μ ≠ λ; (vi) e_{i}M(λ) is irreducible if and only if ε_{i}(λ) = 1.

Hence, res_{H_{n−1}^{fin}}^{H_{n}^{fin}}M(λ) is completely reducible if and only if ε_{i}(λ) ≤ 1 for every i ∈ I.

Proof. The existence of such modules e_{i}M(λ) follows from (6.6), Lemma 6.3 and Theorem 6.6(i), combined as usual with Kang’s Theorem 9.2. Uniqueness follows from Krull-Schmidt and the block classification from Theorem 9.4(iii). For the remaining properties, (iii),(iv) and (v) follow from Theorem 6.11 and Corollary 6.12. Finally, (vi) follows from (iii) as e_{i}M(λ) is a module with simple socle and cosocle both isomorphic to M(λ_{A}).

Theorem 9.6. Let λ ∈ ℛ𝒫 _{h}(n). There exist H_{n+1}^{fin}-modules f_{i}M(λ) for each i ∈ I, unique up to isomorphism, such that

(i) ind_{H_{n}^{fin}}^{H_{n+1}^{fin}}M(λ) ≅ \{ \begin{matrix}2f_{0}M(λ) ⊕ 2f_{1}M(λ) ⊕ ⋯ ⊕ 2f_{ℓ}M(λ) & if b(λ) is odd, \\ f_{0}M(λ) ⊕ 2f_{1}M(λ) ⊕ ⋯ ⊕ 2f_{ℓ}M(λ) & if b(λ) is even; \end{matrix}

(ii) for each i ∈ I, f_{i}M(λ) ≠ 0 if and only if λ has an i-cogood node B, in which case f_{i}M(λ) is a self-dual indecomposable module with irreducible socle and cosocle isomorphic to M(λ^{B}).

Moreover, if i ∈ I and λ has an i-cogood node B, then

(iii) the multiplicity of M(λ^{B}) in f_{i}M(λ) is φ_{i}(λ), φ_{i}(λ^{B}) = φ_{i}(λ) − 1, and φ_{i}(μ) <

φ_{i}(λ) − 1 for all other composition factors M(μ) of f_{i}M(λ); (iv) End_{H_{n+1}^{fin}}(f_{i}M(λ)) ≃ End_{H_{n+1}^{fin}}(M(λ^{B}))^{⊕φ_{i}(λ)} as a vector superspace;

(v) Hom_{H_{n+1}^{fin}}(f_{i}M(λ), f_{i}M(μ)) = 0 for all μ ∈ ℛ𝒫 _{h}(n) with μ ≠ λ; (vi) f_{i}M(λ) is irreducible if and only if φ_{i}(λ) = 1.

Hence, ind_{H_{n}^{fin}}^{H_{n+1}^{fin}}M(λ) is completely reducible if and only if φ_{i}(λ) ≤ 1 for every i ∈ I.

Proof. The argument is the same as Theorem 9.5, but using (6.6), Lemma 6.4, Theorem 6.6(ii), Corollary 6.19 and Theorem 6.26.

There is one H_{n}^{fin}-module that deserves special mention, the so-called basic spin module. Recall from §2-d that the subalgebra H_{n}^{cl} of H_{n}^{fin} generated by T_{1}, …, T_{n−1} is the classical Hecke algebra associated to the symmetric group. It has a one dimensional module denoted 1, on which each T_{i} acts as multiplication by q. For n ≥ 1, we define

I(n) := ind_{H_{n}^{cl}}^{H_{n}^{fin}}1,  (9.11)

giving an H_{n}^{fin}-module of dimension 2^{n}. Also introduce the restricted h-strict partition

ω_{n} := \{ \begin{matrix}(h^{a}, b) & if b ≠ 0, \\ (h^{a−1}, h − 1, 1) & if b = 0, \end{matrix}  (9.12)

where n = ah + b with 0 ≤ b < h. Lemma 9.7. If p ∤ n then I(n) ≅ M(ω_{n}); if p|n then I(n) is an indecomposable module with two composition factors both isomorphic to M(ω_{n}). In particular,

dim M(ω_{n}) = \{ \begin{matrix}2^{n} & if p ∤ n, \\ 2^{n−1} & if p | n. \end{matrix}

Proof. This is obvious if n = 1, 2 and easy to check directly if n = 3. Now for n > 3 we proceed by induction using Theorem 9.5 together with the observation that res_{H_{n−1}^{fin}}^{H_{n}^{fin}}I(n) ≃

I(n − 1) ⊕ ΠI(n − 1). We consider the four cases n ≡ 0, 1 or 2 (mod h) and n ≢ 0, 1, 2 (mod h) separately.

Suppose first that n ≢ 0, 1, 2 (mod h). Considering the crystal graph shows that f̃_{i}ω_{n−1} ≠ 0 only for i = 0 and for one other i ∈ I, for which f̃_{i}ω_{n−1} = ω_{n}. By the induction hypothesis, res_{H_{n−1}^{fin}}^{H_{n}^{fin}}I(n) ≅ 2M(ω_{n−1}). Hence by Theorem 9.5, I(n) can only contain M(ω_{n})

and M(f̃_{0}ω_{n−1}) as composition factors. But the latter case cannot hold since by Theorem 9.5 again, res_{H_{n−1}^{fin}}^{H_{n}^{fin}}M(f̃_{0}ω_{n−1}) is not isotypic. Hence all composition factors of I(n) are ≅ M(ω_{n}),

and one easily gets that in fact I(n) ≅ M(ω_{n}) by a dimension argument.

Next suppose that n ≡ 0 (mod h). This time, f̃_{0}ω_{n−1} = ω_{n} and all other f̃_{i}ω_{n−1} are zero. Hence, by the induction hypothesis and the branching rules, I(n) only involves M(ω_{n}) as a constituent. But we have that res_{H_{n−1}^{fin}}^{H_{n}^{fin}}M(ω_{n}) = e_{0}M(ω_{n}) ≅ M(ω_{n−1}) so in fact that

I(n) must have M(ω_{n}) as a constituent with multiplicty two. Further consideration of the endomorphism ring of I(n) shows moreover that it is an indecomposable module.

The argument in the remaining two cases n ≡ 1 (mod h) and n ≡ 2 (mod h) is entirely similar.

§9-c. Projective representations of S_{n}. We specialize for the final applications to the degenerate case. So now F is an algebraically closed field of characteristic p ≠ 2, h = p (if p ≠ 0) or h = ∞ (if p = 0), and ℓ = (h − 1)/2. We are interested in the projective representations of the symmetric group S_{n} over the field F. Equivalently, see for example [BK2, §3], we consider the representations of the twisted group algebra S(n), defined by generators t_{1}, …, t_{n−1} subject to the relations

t_{i}^{2} = 1,  t_{i}t_{i+1}t_{i} = t_{i+1}t_{i}t_{i+1},  t_{i}t_{j} = −t_{j}t_{i}

for all 1 ≤ i ≤ n − 1 and all 1 ≤ j ≤ n − 1 with |i − j| > 1. We view S(n) as a superalgebra, defining the grading by declaring the generators t_{1}, …, t_{n−1} to be of degree ¯1. All modules in this subsection will be ℤ_{2}-graded as usual, see §2-b.

Let W (n) denote the finite Sergeev superalgebra, replacing the notation H_{n}^{fin} used previously. Recall from §2-k that W (n) is a twisted tensor product of the group algebra F S_{n} of the symmetric group and the Clifford superalgebra C(n). So W (n) has even generators s_{1}, …, s_{n−1} subject to the usual relations of the basic transpositions in S_{n}, odd generators c_{1}, …, c_{n} subject to the Clifford relations as in (2.7), (2.8), and the additional relations (2.35). The connection between the superalgebras S(n) and W (n) is explained by the following observation, due originally to Sergeev [S2], see also [BK2, 3.3]:

Lemma 9.8. There is an isomorphism of superalgebras

φ : S(n) ⊗ C(n) ⟶^{∼} W (n)

such that φ(1 ⊗ c_{i}) = c_{i} and φ(t_{j} ⊗ 1) = \frac{1}{\sqrt{−2}}s_{j}(c_{j} − c_{j+1}), i = 1, …, n, j = 1, …, n − 1.

We will from now on identify W (n) with S(n) ⊗ C(n) according to the lemma, and view S(n) (resp. C(n)) as the subalgebra S(n) ⊗ 1 (resp. 1 ⊗ C(n)) of W (n). Recall the antiautomorphism τ of W (n) defined as in (2.39) to be the unique antiautomorphism which is the identity on the generators s_{1}, …, s_{n−1}, c_{1}, …, c_{n}. This leaves invariant the subalgebra S(n) of W (n), so induces an antiautomorphism

τ : S(n) → S(n).  (9.13)

More explicitly, τ on S(n) is the unique antiautomorphism which maps the generator t_{i} to −t_{i} for each i = 1, …, n − 1. Given a finite dimensional S(n)- (resp. W (n)-) module M we write M^{τ} for the dual vector superspace viewed as a module by twisting the natural right action into a left action via the antiautomorphism τ, see §2-b.

We also need the Clifford module U(n), which is the unique irreducible C(n)-module up to isomorphism, see for instance [BK2, 2.10]. We recall that U(n) is of type M if n is even, type Q if n is odd, and dim U(n) = 2^{⌊(n+1)/2⌋}.

Now consider the exact functors

F_{n} : Rep S(n) → Rep W (n),  F_{n} :=? ⊠ U(n),
G_{n} : Rep W (n) → Rep S(n),  G_{n} := Hom_{C(n)}(U(n),?).

Also let

res_{W (n−1)}^{W (n)} : Rep W (n) → Rep W (n − 1), ind_{W (n−1)}^{W (n)} : Rep W (n − 1) → Rep W (n),
res_{S(n−1)}^{S(n)} : Rep S(n) → Rep S(n − 1), ind_{S(n−1)}^{S(n)} : Rep S(n − 1) → Rep S(n)

denote the (exact) induction and restriction functors, where S(n−1) ⊂ S(n) and W (n−1) ⊂ W (n) are the natural subalgebras generated by all but the last generators. Recalling that Π denotes the parity change functor (2.1), the following lemma lists some basic properties:

Lemma 9.9. The functors F_{n} and G_{n} are left and right adjoint to one another, and both commute with τ-duality. Moreover:

(i) Suppose that n is even. Then F_{n} and G_{n} are inverse equivalences of categories, so induce a type-preserving bijection between the isomorphism classes of irreducible S(n)-modules and of irreducible W (n)-modules. Also,

F_{n−1} ∘ res_{S(n−1)}^{S(n)} ≃ res_{W (n−1)}^{W (n)} ∘ F_{n},  (9.14)
G_{n−1} ∘ res_{W (n−1)}^{W (n)} ≃ res_{S(n−1)}^{S(n)} ∘ G_{n} ⊕ Π ∘ res_{S(n−1)}^{S(n)} ∘ G_{n},  (9.15)
F_{n+1} ∘ ind_{S(n)}^{S(n+1)} ≃ ind_{W (n)}^{W (n+1)} ∘ F_{n},  (9.16)
G_{n+1} ∘ ind_{W (n)}^{W (n+1)} ≃ ind_{S(n)}^{S(n+1)} ∘ G_{n} ⊕ Π ∘ ind_{S(n)}^{S(n+1)} ∘ G_{n}.  (9.17)

(ii) Suppose that n is odd. Then F_{n} ∘ G_{n} ≃ Id ⊕Π and G_{n} ∘ F_{n} ≃ Id ⊕Π. Furthermore, the functor F_{n} induces a bijection between isomorphism classes of irreducible S(n)- modules of type M and irreducible W (n)-modules of type Q, while the functor G_{n} induces a bijection between isomorphism classes of irreducible W (n)-modules of type M and irreducible S(n)-modules of type Q. Finally,

res_{W (n−1)}^{W (n)} ∘ F_{n} ≃ F_{n−1} ∘ res_{S(n−1)}^{S(n)} ⊕ Π ∘ F_{n−1} ∘ res_{S(n−1)}^{S(n)},  (9.18)
res_{S(n−1)}^{S(n)} ∘ G_{n} ≃ G_{n−1} ∘ res_{W (n−1)}^{W (n)},  (9.19)
ind_{W (n)}^{W (n+1)} ∘ F_{n} ≃ F_{n+1} ∘ ind_{S(n)}^{S(n+1)} ⊕ Π ∘ F_{n+1} ∘ ind_{S(n)}^{S(n+1)},  (9.20)
ind_{S(n)}^{S(n+1)} ∘ G_{n} ≃ G_{n+1} ∘ ind_{W (n)}^{W (n+1)}.  (9.21)

Proof. Most of these facts are proved in [BK2, 3.4,3.5], but we recall some of the details since we need to go slightly further. Let us consider the proof that F_{n} ∘ G_{n} ≃ Id ⊕Π and G_{n} ∘ F_{n} ≃ Id ⊕Π, assuming that n is odd. Let I, J be a basis for End_{C(n)}(U(n)) with I being the identity and J being an odd involution. Then, there are natural isomorphisms

η : F_{n} ∘ G_{n} ⟶^{∼} Id ⊕Π, ξ : Id ⊕Π ⟶^{∼} G_{n} ∘ F_{n}.

The first is defined for each W (n)-module M by η_{M} : Hom_{C(n)}(U(n), M)⊠U(n) → M ⊕ΠM, θ ⊗ u ↦ (θ(u), (−1)θ̄θ(Ju)). The second is defined for each S(n)-module N by ξ_{N} : N ⊕ ΠN → Hom_{C(n)}(U(n), N ⊠U(n)), (n, n′) ↦ θ_{n,n′}, where θ_{n,n′}(u) = n⊗u+(−1)^{n̄′}n′ ⊗Ju for each u ∈ U(n). The proof that η and ξ really are isomorphisms is similar to the argument in [BK2, 3.4]. Now consider the composite natural transformations

Id ⟶ Id ⊕Π ⟶^{ξ} G_{n} ∘ F_{n},  F_{n} ∘ G_{n} ⟶^{η} Id ⊕Π ⟶ Id,  (9.22)
Id ⟶ Id ⊕Π ⟶^{η^{−1}} F_{n} ∘ G_{n},  G_{n} ∘ F_{n} ⟶^{ξ^{−1}} Id ⊕Π ⟶ Id,  (9.23)

where the unmarked arrows are the obvious even ones. Then, (9.22) (resp. 9.23) gives the unit and counit of the adjunction needed to prove that F_{n} is left (resp. right) adoint to G_{n}. We leave the details to be checked to the reader, see e.g. [ML, IV.1, Theorem 2(v)].

In case n is even, a similar but easier argument shows that F_{n} ∘ G_{n} ≃ Id, G_{n} ∘ F_{n} ≃ Id so that F_{n} and G_{n} are inverse equivalences, hence left and right adjoint to each other. Now the statements in (i) and (ii) about isomorphism classes of irreducible modules follow easily as in [BK2, 3.5].

Let us next prove that F_{n} commutes with duality. The antiautomorphism τ of W (n) induces the antiautomorphism τ of the subalgebra C(n) with τ(c_{i}) = c_{i} for each i = 1, …, n. Let φ : U(n) → U(n)^{τ} , u ↦ φ_{u} be a homogeneous isomorphism; note that it is not always possible to choose φ to be even, since U(n) ≄ U(n)^{τ} in case n ≡ 2 (mod 4). We get a natural isomorphism

Φ_{M} : M^{τ} ⊠ U(n) → (M ⊠ U(n))^{τ} , f ⊗ u ↦ θ_{f,u}

for each finite dimensional S(n)-module M, where θ_{f,u}(m⊗v) = (−1)^{m̄φ̄}f(m)φ_{u}(v) for each m ∈ M, v ∈ U(n). This shows that F_{n} commutes with duality, i.e. τ ∘ F_{n} ∘ τ ≅ F_{n}. Hence, using (2.2) and the fact that G_{n} is left adjoint to F_{n}, the composite functor τ ∘ G_{n} ∘ τ is right adjoint to F_{n}. But we already know that G_{n} is right adjoint to F_{n}, so uniqueness of adjoints gives that τ ∘ G_{n} ∘ τ ≅ G_{n}. This shows that G_{n} commutes with duality too.

It just remains to check the isomorphisms (9.14)–(9.21). Well, (9.14) follows from the definition on noting that res_{C(n−1)}^{C(n)}U(n) ≃ U(n − 1) if n is even. Then (9.15) follows from (9.14) on composing on the left with G_{n−1} and on the right with G_{n}. Next, (9.19) follows from the definition and an application of Frobenius reciprocity, using the observation that U(n) ≃ ind_{C(n−1)}^{C(n)}U(n − 1) if n is odd. As before (9.18) then follows, composing with F_{n−1} and F_{n}. Finally, (9.16), (9.17), (9.20) and (9.21) follow from (9.19), (9.18), (9.15) and (9.14) respectively by uniqueness of adjoints.

We are ready to derive the consequences for S(n) of the results of §9-b, or rather, of the analogous results for W (n) in the degenerate case. By Theorem 9.4, we have a parametrization {M(λ) | λ ∈ ℛ𝒫 _{p}(n)} of the irreducible W (n)-modules. Lemma 9.9 shows that the functors F_{n} and G_{n} set up a natural correspondence between classes of irreducible S(n) and W (n)-modules, type-preserving if n is even and type-reversing if n is odd. Hence we have a parametrization {D(λ) | λ ∈ ℛ𝒫 _{p}(n)} of the irreducible S(n)-modules, letting D(λ) be an irreducible S(n)-module corresponding to M(λ) under the correspondence. Also, recalling the definition (9.9), define

a(λ) := n − b(λ)  (9.24)

for λ ∈ ℛ𝒫 _{p}(n). We observe by (9.10) that

a(λ) ≡ γ_{1} + ⋯ + γ_{ℓ} (mod 2),  (9.25)

where γ_{1} + ⋯ + γ_{ℓ} counts the number of nodes in the Young diagram λ of residue different from 0. Then:

Theorem 9.10. The modules {D(λ) | λ ∈ ℛ𝒫 _{p}(n)} form a complete set of pairwise nonisomorphic irreducible S(n)-modules. Moreover, for λ, μ ∈ ℛ𝒫 _{p}(n),

(i) D(λ) ≅ D(λ)^{τ}; (ii) D(λ) is of type M if a(λ) is even, type Q if a(λ) is odd; (iii) D(μ) and D(λ) belong to the same block if and only if cont(μ) = cont(λ); (iv) D(λ) is a projective module if and only if λ is a p-bar core.

Proof. It just remains to observe that (i)–(iv) follow follow directly from Theorem 9.4(i)– (iv) using Lemma 9.9.

Remark 9.11. The p-blocks of the ordinary irreducible projective representations of S_{n} were described by Humphreys [H], in terms of the notion of p-bar core. However, unlike the case of S_{n}, Humphreys’ result does not imply Theorem 9.10(iii) because of the lack of information on decomposition numbers.

Define ω_{n} ∈ ℛ𝒫 _{p}(n) as in (9.12). Then the irreducible S(n)-module D(ω_{n}) is the basic spin module:

Lemma 9.12. D(ω_{n}) is of dimension 2^{⌊n/2⌋}, unless p|n when its dimension is 2^{⌊(n−1)/2⌋}. Moreover, D(ω_{n}) is equal to the reduction modulo p of the basic spin module D((n))_{ℂ} of S(n)_{ℂ} over ℂ, except if p|n and n is even when the reduction modulo p of D((n))_{ℂ} has two composition factors both isomorphic to D(ω_{n}). Proof. The statement about dimension is immediate from Lemmas 9.7 and 9.9. The final statement is easily proved by working in terms of W (n) and using the explicit construction given in (9.11).

To motivate the next two theorems, note that the map [D(λ)] ↦ [M(λ)] for each λ ∈ ℛ𝒫 _{p}(n) extends linearly to an isomorphism

K(Rep S(n)) ⟶^{∼} K(Rep W (n))

at the level of Grothendieck groups. Using this identification, we can lift the operators e_{i} and f_{i} on K(Λ_{0}) = ⨁_{n≥0} K(Rep W (n)) defined earlier to define similar operators on ⨁_{n≥0} K(Rep S(n)). Then all our earlier results about K(Λ_{0}), for instance Theorems 7.15 and 7.16, could be restated purely in terms of the representations of S(n) instead of W (n). In fact, we can do slightly better and define the operators e_{i} and f_{i} on irreducible S(n)- modules, not just on the Grothendieck group:

Theorem 9.13. Let λ ∈ ℛ𝒫 _{p}(n). There exist S(n − 1)-modules e_{i}D(λ) for each i ∈ I, unique up to isomorphism, such that

(i) res_{S(n−1)}^{S(n)}D(λ) ≅ \{ \begin{matrix}e_{0}D(λ) ⊕ 2e_{1}D(λ) ⊕ ⋯ ⊕ 2e_{ℓ}D(λ) & if a(λ) is odd, \\ e_{0}D(λ) ⊕ e_{1}D(λ) ⊕ ⋯ ⊕ e_{ℓ}D(λ) & if a(λ) is even; \end{matrix}

(ii) for each i ∈ I, e_{i}D(λ) ≠ 0 if and only if λ has an i-good node A, in which case e_{i}D(λ) is a self-dual indecomposable module with irreducible socle and cosocle isomorphic to D(λ_{A}).

Moreover, if i ∈ I and λ has an i-good node A, then

(iii) the multiplicity of D(λ_{A}) in e_{i}D(λ) is ε_{i}(λ), ε_{i}(λ_{A}) = ε_{i}(λ)−1, and ε_{i}(μ) < ε_{i}(λ)−1

for all other composition factors D(μ) of e_{i}D(λ); (iv) End_{S(n−1)}(e_{i}D(λ)) ≃ End_{S(n−1)}(D(λ_{A}))^{⊕ε_{i}(λ)} as a vector superspace;

(v) Hom_{S(n−1)}(e_{i}D(λ), e_{i}D(μ)) = 0 for all μ ∈ ℛ𝒫 _{p}(n) with μ ≠ λ; (vi) e_{i}D(λ) is irreducible if and only if ε_{i}(λ) = 1.

Hence, res_{S(n−1)}^{S(n)}D(λ) is completely reducible if and only if ε_{i}(λ) ≤ 1 for every i ∈ I.

Proof. If n is odd, we simply define e_{i}D(λ) := G_{n−1}(e_{i}M(λ)) for each i ∈ I, λ ∈ ℛ𝒫 _{p}(n). If n is even, take

e_{i}D(λ) := \{ \begin{matrix}G_{n−1}(e_{i}M(λ)) & if a(λ) is even and i ≠ 0, or a(λ) is odd and i = 0, \\ \overline{G}_{n−1}(e_{i}M(λ)) & if a(λ) is even and i = 0, or a(λ) is odd and i ≠ 0. \end{matrix}

We need to explain the notation \overline{G}_{n−1} used in the last two cases: here, e_{i}M(λ) admits an odd involution by Remark 6.7 and Theorem 9.4(ii), and also U(n − 1) has an odd involution since n is even. So in exactly the same way as in the definition of (6.7), we can introduce the space

\overline{G}_{n−1}(e_{i}M(λ)) := \overline{Hom}_{C(n−1)}(U(n − 1), e_{i}M(λ)).

It is then the case that G_{n−1}(e_{i}M(λ)) ≃ \overline{G}_{n−1}(e_{i}M(λ)) ⊕ Π\overline{G}_{n−1}(e_{i}M(λ)). Equivalently, by Lemma 9.9(ii) e_{i}D(λ) can be characterized by

e_{i}M(λ) ≅ F_{n−1}(e_{i}D(λ))

if a(λ) is even and i = 0, or a(λ) is odd and i ≠ 0.

With these definitions, it is now a straightforward matter to prove (i)–(vi) using Theorem 9.5 and Lemma 9.9. Finally, the uniqueness statement is immediate from Krull-Schmidt and the description of blocks from Theorem 9.10(iii).

Theorem 9.14. Let λ ∈ ℛ𝒫 _{p}(n). There exist S(n + 1)-modules f_{i}D(λ) for each i ∈ I, unique up to isomorphism, such that

(i) ind_{S(n)}^{S(n+1)}D(λ) ≅ \{ \begin{matrix}f_{0}D(λ) ⊕ 2f_{1}D(λ) ⊕ ⋯ ⊕ 2f_{ℓ}D(λ) & if a(λ) is odd, \\ f_{0}D(λ) ⊕ f_{1}D(λ) ⊕ ⋯ ⊕ f_{ℓ}D(λ) & if a(λ) is even; \end{matrix}

(ii) for each i ∈ I, f_{i}D(λ) ≠ 0 if and only if λ has an i-cogood node B, in which case f_{i}D(λ) is a self-dual indecomposable module with irreducible socle and cosocle isomorphic to D(λ^{B}).

Moreover, if i ∈ I and λ has an i-cogood node B, then

(iii) the multiplicity of D(λ^{B}) in f_{i}D(λ) is φ_{i}(λ), φ_{i}(λ^{B}) = φ_{i}(λ) − 1, and φ_{i}(μ) < φ_{i}(λ) − 1 for all other composition factors D(μ) of f_{i}D(λ); (iv) End_{S(n+1)}(f_{i}D(λ)) ≃ End_{S(n+1)}(D(λ^{B}))^{⊕φ_{i}(λ)} as a vector superspace;

(v) Hom_{S(n+1)}(f_{i}D(λ), f_{i}D(μ)) = 0 for all μ ∈ ℛ𝒫 _{p}(n) with μ ≠ λ; (vi) f_{i}D(λ) is irreducible if and only if φ_{i}(λ) = 1.

Hence, ind_{S(n)}^{S(n+1)}M(λ) is completely reducible if and only if φ_{i}(λ) ≤ 1 for every i ∈ I. Proof. This is deduced from Theorem 9.6 by similar argument to the proof of Theorem 9.13.

Remark 9.15. (i) In [BK2, 10.3], we gave an entirely different construction of the irreducible S(n)-modules, which we also denoted by D(λ) for λ ∈ ℛ𝒫 _{p}(n). We warn the reader that we have not yet proved that the modules denoted D(λ) here are isomorphic to those in [BK2], though we expect this to be the case.

(ii) Over ℂ, the branching rules in the preceeding two theorems are the same as Morris’ branching rules, see [Mo]. In particular using this observation, one easily shows that our labelling of irreducibles over ℂ agrees with the standard labelling. Hence over ℂ the labelling here agrees with the labelling in [BK2], compare [S1].

(iii) There is one other case where it is easy to see right away that the labelling here agrees with [BK2]: if λ is a p-bar core then consideration of central characters shows that D(λ) is equal to a reduction modulo p of the irreducible representation of S(n)_{ℂ} over ℂ with the same label. So it coincides with the module D(λ) of [BK2] thanks to [BK2, 10.8].

§9-d. The Jantzen-Seitz problem. The results of the previous subsection give a solution to the Jantzen-Seitz problem for projective representations of the symmetric and alternating groups. This problem originated in [JS], and is of interest in the study of maximal subgroups of the finite classical groups. To consider the Jantzen-Seitz problem, we first need to switch to studying ungraded representations of the twisted group algebra S(n). The goal is to describe all ungraded irreducible S(n)-modules which remain irreducible on restriction to the subalgebra S(n − 1).

As in [BK2, §10], it is straightforward to obtain a parametrization of the ungraded irreducible S(n)-modules from Theorem 9.10: if a(λ) is odd then D(λ) decomposes as an ungraded module as D(λ) = D(λ, +) ⊕ D(λ, −) for two non-isomorphic irreducible S(n)- modules D(λ, +) and D(λ, −). If a(λ) is even, then D(λ) is irreducible viewed as an ungraded S(n)-module, but we denote it instead by D(λ, 0) to make it clear that we are no longer considering a ℤ_{2}-grading. Then

{D(λ, 0) | λ ∈ ℛ𝒫 _{p}(n), a(λ) even} ⊔ {D(λ, +), D(λ, −) | λ ∈ ℛ𝒫 _{p}(n), a(λ) odd}

gives a complete set of pairwise non-isomorphic ungraded irreducible S(n)-modules.

Remark 9.16. Using Theorem 9.10 and a counting argument involving Humphreys’ block classification [H], it is not hard to obtain the following description of the ungraded blocks of the algebra S(n). Let D(λ, ε) and D(μ, δ) be ungraded irreducible S(n)-modules. Then, with one exception, D(λ, ε) and D(μ, δ) lie in the same block if and only if λ and μ have the same p-bar core. The exception is if λ = μ is a p-bar core, a(λ) is odd and ε = −δ, when D(λ, ε) and D(μ, δ) are in different blocks.

Now we state the solution to the Jantzen-Seitz problem for projective representations of the symmetric group. The proof is a straightforward consequence of Theorem 9.13.

Theorem 9.17. Let λ ∈ ℛ𝒫 _{p}(n). Then:

(i) If a(λ) is even, res_{S(n−1)}^{S(n)}D(λ, 0) is irreducible if and only if ε_{0}(λ) = ∑_{i∈I} ε_{i}(λ) = 1.

(ii) If a(λ) is odd, res_{S(n−1)}^{S(n)}D(λ, ±) is irreducible if and only if ∑_{i∈I} ε_{i}(λ) = 1.

Finally let us discuss the analogous problem for the projective representations of the alternating group. As explained in [BK2, §10], it suffices for this to consider the representation theory of the algebra A(n) := S(n)¯0, i.e. the twisted group algebra of the alternating group. The irreducible A(n)-modules (there being no ambiguity between graded and ungraded modules since A(n) is purely even) are constructed out of those for S(n) as in [BK2, Theorem 10.4]. More precisely, if λ ∈ ℛ𝒫 _{p}(n) has a(λ) even, then res_{A(n)}^{S(n)}D(λ) decomposes as a direct sum E(λ, +) ⊕ E(λ, −) of two non-isomorphic irreducible A(n)-modules. If a(λ) is odd, then res_{A(n)}^{S(n)}D(λ) decomposes as a direct sum of two copies of a single irreducible A(n)-module denoted E(λ, 0). Then,

{E(λ, 0) | λ ∈ ℛ𝒫 _{p}(n), a(λ) odd} ⊔ {E(λ, +), E(λ, −) | λ ∈ ℛ𝒫 _{p}(n), a(λ) even}

gives a complete set of pairwise non-isomorphic irreducible A(n)-modules. Now the solution to the Jantzen-Seitz problem in this case, again an easy consequence of Theorem 9.13, is as follows:

Theorem 9.18. Let λ ∈ ℛ𝒫 _{p}(n). Then:

(i) If a(λ) is even, res_{A(n−1)}^{A(n)}E(λ, ±) is irreducible if and only if ∑_{i∈I} ε_{i}(λ) = 1;

(ii) If a(λ) is odd, res_{A(n−1)}^{A(n)}E(λ, 0) is irreducible if and only if ε_{0}(λ) = ∑_{i∈I} ε_{i}(λ) = 1.

References

[AF] F. Anderson and K. Fuller, Rings and categories of modules, Springer-Verlag, 1974.

[A1] S. Ariki, On the decomposition numbers of the Hecke algebra of type G(m, 1, n), J. Math. Kyoto Univ. 36 (1996), 789–808.

[A2] On the classification of simple modules for cyclotomic Hecke algebras of type G(m, 1, n) and Kleshchev multipartitions, to appear in Osaka J. Math..

[AM] S. Ariki and A. Mathas, The number of simple modules of the Hecke algebras of type G(r, 1, n), Math. Z. 233 (2000), 601-623.

[Be] G. Bergman, The diamond lemma for ring theory, Advances Math. 29 (1978), 178–218.

[B] J. Brundan, Modular branching rules and the Mullineux map for Hecke algebras of type A, Proc. London Math. Soc. 77 (1998), 551–581.

[BK1] J. Brundan and A. Kleshchev, Translation functors for general linear and symmetric groups, Proc. London Math. Soc. 80 (2000), 75–106.

[BK2] J. Brundan and A. Kleshchev, Projective representations of the symmetric group via Sergeev duality, to appear in Math. Z..

[DJ] R. Dipper and G. James, Representations of Hecke algebras of general linear groups. Proc. London Math. Soc. 52 (1986), 20–52.

[G1] I. Grojnowski, Affine 𝔰̂𝔩_{p} controls the modular representation theory of the symmetric group and related Hecke algebras, preprint, 1999.

[G2] I. Grojnowski, Blocks of the cyclotomic Hecke algebra, preprint, 1999.

[GV] I. Grojnowski and M. Vazirani, Strong multiplicity one theorem for affine Hecke algebras of type A, Transf. Groups., to appear.

[H] J. F. Humphreys, Blocks of projective representations of the symmetric groups, J. London Math. Soc. 33 (1986), 441–452.

[JS] J. C. Jantzen and G. M. Seitz, On the representation theory of the symmetric groups, Proc. London Math. Soc. 65 (1992), 475–504.

[JN] A. Jones and M. Nazarov, Affine Sergeev algebra and q-analogues of the Young symmetrizers for projective representations of the symmetric group, Proc. London Math. Soc. 78 (1999), 481–512.

[Kc] V. Kac, Infinite dimensional Lie algebras, Cambridge University Press, third edition, 1995.

[Kan] S.-J. Kang, Crystal bases for quantum affine algebras and combinatorics of Young walls, preprint, Seoul National University, 2000.

[Ka] M. Kashiwara, “On crystal bases”, in: Representations of groups (Banff 1994), CMS Conf. Proc. 16 (1995), 155–197.

[KS] M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), 9–36.

[K1] A. Kleshchev, Branching rules for modular representations of symmetric groups II, J. reine angew. Math. 459 (1995), 163–212.

[K2] A. Kleshchev, Branching rules for modular representations of symmetric groups III: some corollaries and a problem of Mullineux, J. London Math. Soc. 54 (1996), 25–38.

[K3] A. Kleshchev, On decomposition numbers and branching coefficients for symmetric and special linear groups, Proc. London Math. Soc. 75 (1997), 497–558.

[LT] B. Leclerc and J.-Y. Thibon, q-Deformed Fock spaces and modular representations of spin symmetric groups, J. Phys. A 30 (1997), 6163–6176.

[Le] D.A. Leites, Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1980), 1–64.

[ML] S. MacLane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, 1971.

[Man] Yu I. Manin, Gauge field theory and complex geometry, Grundlehren der mathematischen Wissenschaften 289, second edition, Springer, 1997.

[Mo] A.O. Morris, The spin representations of the symmetric group, Canad. J. Math. 17 (1965), 543-549.

[N] M. Nazarov, Young’s symmetrizers for projective representations of the symmetric group, Advances Math. 127 (1997), 190–257.

[O] G. Olshanski, Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra, Lett. Math. Phys. 24 (1992), 93–102.

[S1] A. N. Sergeev, Tensor algebra of the identity representation as a module over the Lie superalgebras GL(n, m) and Q(n), Math. USSR Sbornik 51 (1985), 419–427.

[S2] A. N. Sergeev, The Howe duality and the projective representations of symmetric groups, Represent. Theory 3 (1999), 416–434.

[V] M. Vazirani, Filtrations on the Mackey decomposition for cyclotomic Hecke algebras, preprint, 1999.

Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

   E-mail address: brundan@darkwing.uoregon.edu, klesh@math.uoregon.edu