The number of simple modules of the Hecke algebras of type G(r, 1, n)

Susumu Ariki and Andrew Mathas

Abstract

This paper classifies the simple modules of the cyclotomic Hecke algebras of type G(r, 1, n) and the affine Hecke algebras of type A in arbitrary characteristic. We do this by first showing that the simple modules of the cyclotomic Hecke algebras are indexed by the set of “Kleshchev multipartitions”.

Introduction

Let n and r be integers with n ≥ 0 and r ≥ 1. Let R be a commutative ring with 1 and let q, Q_{1}, …, Q_{r} be elements of R with q invertible. The cyclotomic Hecke algebra ℋ_{R,n} = ℋ_{R,n} (q; {Q_{1}, …, Q_{r}}) of type G(r, 1, n) is the unital associative R–algebra with generators T_{0}, T_{1}, …, T_{n−1} and relations

(T_{0} − Q_{1}) ⋯ (T_{0} − Q_{r}) = 0,
T_{0}T_{1}T_{0}T_{1} = T_{1}T_{0}T_{1}T_{0},
(T_{i} + 1)(T_{i} − q) = 0  for 1 ≤ i ≤ n − 1,
T_{i+1}T_{i}T_{i+1} = T_{i}T_{i+1}T_{i} for 1 ≤ i ≤ n − 2,
T_{i}T_{j} = T_{j}T_{i}  for 0 ≤ i < j − 1 ≤ n − 2.

The first result of this paper says that, in a certain sense, the number of simple ℋ_{R,n} –modules is independent of the base field and independent of q. To state this precisely, let ℓ be the smallest positive integer such that 1 + q + ⋯ + q^{ℓ−1} = 0; set ℓ = ∞ if no such integer exists.

Theorem A Suppose that R is a field and let {Q_{1}, …, Q_{r}} = S_{0} ⊔S_{1} ⊔…⊔S_{a} be a partition of the parameter set such that S_{0} = { Q_{i} | Q_{i} = 0 } and, for the remaining non–zero parameters, Q_{i}/Q_{j} is a power of q if and only if there exists an integer b such that Q_{i} and Q_{j} both belong to S_{b}. For i = 1, 2, …, a choose s_{i} ∈ S_{i} and define n_{ij} to be the multiplicity of s_{i}q^{j} in S_{i}. Then the number of simple ℋ_{R,n} –modules depends only upon ℓ and the integers n_{ij}.

A.M.S subject classification (1991): 16G99, 17B67, 20G05.

The first author was supported in part by the DFG–JSPS program ”Representation Theory of Finite and Algebraic Groups”

When q = 1 the simple ℋ_{R,n} –modules have been classified in [22] and it is easy to see that Theorem A is true in this case. When q ≠ 1 the arguments of [24,27] allow us to reduce to the case where all of Q_{i} are non–zero and Q_{i}/Q_{j} is always a power of q. The key idea now is to vary n and consider the (graded dual of the) direct sum of the Grothendieck groups of ℋ_{R,n} ; by [1], this is an irreducible module of a Kac–Moody algebra when R is a field of characteristic zero. In fact, this is true independently of the characteristic of R, from which the result follows.

To date, few studies have been made of the modular representations of affine Hecke algebras in fields of positive characteristic. Theorem A, combined with results of Chriss and Ginzburg [5], yields a classification of the irreducible representations of the affine Hecke algebras of type A over arbitrary fields. Let Ĥ_{R,n} be the affine Hecke algebra associated to GL(n, F) where F is a local field; the algebra Ĥ_{ℤ[q,q^{−1}],n} is again a ℤ[q, q^{−1}]–algebra.

A segment of length d ≥ 1 is a sequence of d consecutive residues [i, i + 1, …, i + d − 1] where i, i + 1, …, i + d − 1 ∈ ℤ/ℓℤ. A multisegment m is an unordered collection of segments; we denote by |m| the sum of the lengths of the segments in m. A multisegment is aperiodic if for every d there exists an i ∈ ℤ/ℓℤ such that [i, i + 1, …, i + d − 1] does not appear in m. Denote by M_{ℓ} the set of aperiodic multisegments and let k_{q}^{×} = k^{×}/<q>. For any field k, let

M_{ℓ}^{n}(k) = \{ \underline{λ}: k_{q}^{×} ⟶M_{ℓ} | ∑_{x∈k_{q}^{×}} |\underline{λ}(x)| = n \} .

Then, using Theorem A, we can deduce the following result.

Theorem B Suppose that k is a field and that q ≠ 1 has order ℓ in k. Let (K, O, k) be a modular system such that q lifts to an element of order ℓ in O.

(i) The simple Ĥ_{k,n}–modules are indexed by M_{ℓ}^{n}(k). (ii) There is a set of Ĥ_{O,n}–modules \{ S\underline{λ} | \underline{λ} ∈ M_{ℓ}^{n}(k) \} such that

(a) each S\underline{λ} ⊗ K is a simple Ĥ_{K,n}–module; and,

(b) S\underline{λ} ⊗k has a simple head and the irreducible quotients form a complete set of simple Ĥ_{k,n}–modules.

As explained in section 3, by work of Vigneras this result completes the classification of the irreducible admissible representations of the general linear groups defined over p–adic fields.

Returning now to the simple ℋ_{R,n} –modules, integrable representations of Kac–Moody algebras have a basis indexed by the vertices of Kashiwara’s crystal graph; by determining this graph we obtain our next result.

Theorem C Suppose that R is a field, q ≠ 1 and that Q_{1}, …, Q_{r} are all non– zero elements of R. Then the irreducible ℋ_{R,n} –modules are indexed by the set of Kleshchev multipartitions.

We define Kleshchev multipartitions below. We remark that we could include the case where some of the Q_{i} are 0 in Theorem C; however, this would overly complicate the definition of Kleshchev multipartitions.

Before giving the definition of Kleshchev multipartitions, we note the following consequence of Theorem C and the Weyl–Kac q–dimension formula. To do this we introduce formal power series F_{ℓ}({n_{i}}_{i∈ℤ/ℓℤ}; x), when ℓ is finite, and F_{∞}({n_{i}}_{i∈ℤ}; x), when ℓ is infinite.

We treat the case ℓ < ∞ first. Suppose that n_{0}, …, n_{ℓ−1} are integers and define F_{ℓ}({n_{i}}_{i∈ℤ/ℓℤ}; x) to be the formal power series

∏_{1≤i<j≤ℓ} \frac{1 − x^{N_{ji}}}{1 − x^{j−i}} ∏_{k>0} (\frac{1 − x^{kN_{ℓ}}}{1 − x^{kℓ}} )^{ℓ−1} ∏_{1≤i<j≤ℓ} \frac{1 − (x^{N_{ij}} + x^{N_{ji}})x^{kN_{ℓ}} + x^{2kN_{ℓ}}}{1 − (x^{i−j} + x^{j−i})x^{kℓ} + x^{2kℓ}}

where N_{i} = ∑_{j=0}^{i−1}(n_{j} + 1) and N_{ij} = N_{i} − N_{j}. This is invariant under a cyclic permutation of the indices.

Suppose next that ℓ = ∞. Then, given integers …, n_{−1}, n_{0}, n_{1}, … such that only finitely many of them are non–zero, let F_{∞}({n_{i}}_{i∈ℤ}; x) be the rational function

∏_{i<j} \frac{1 − x^{N_{ji}}}{1 − x^{j−i}}

where N_{ji} = ∑_{i≤k<j}(n_{k} + 1). This is invariant under the shift operation on the indices.

Finally, we define F_{ℓ}(x) = F_{ℓ}({n_{i}}; x) where n_{i} = δ_{i0} for all i ∈ ℤ/ℓℤ.

The following theorem gives the precise number of Kleshchev diagrams associated with a parameter set {q, Q_{1}, …, Q_{r}}; as we explain in the final section, this has important consequences for the classification of the irreducible modules of the finite groups of Lie type in non–defining characteristic.

Theorem D Suppose that R is a field, q ≠ 1 and define integers n_{ij} relative to a partitioning of the parameter set {Q_{1}, …, Q_{r}} = S_{0} ⊔ S_{1} ⊔ … ⊔ S_{a} as in Theorem A. Then the generating function for the number of simple ℋ_{R,n} – modules is

F_{ℓ}(x) ∏_{i=1}^{a} F_{ℓ}({n_{ij}}_{j∈ℤ/ℓℤ}; x).

By Theorem 1.6, the simple modules of ℋ_{R,n} (q; S_{0}) are indexed by multipartitions of the form ((0), …, (0), μ) where μ is an ℓ–restricted partition; so, in fact, F_{ℓ}(x) is the generating function for the set of ℓ–restricted partitions.

For example,

F_{3}(x) = ∏_{k>0} \frac{(1 − x^{4k−2})(1 − x^{4k−1})^{2}(1 − x^{4k})^{2}(1 − x^{4k+1})^{2}(1 − x^{4k+2})}{(1 − x^{3k−2})(1 − x^{3k−1})^{2}(1 − x^{3k})^{2}(1 − x^{3k+1})^{2}(1 − x^{3k+2})}

= \frac{1}{(1 − x)} ∏_{k>0} \frac{(1 − x^{4k−1})(1 − x^{4k})(1 − x^{4k+1})(1 − x^{4k+2})}{(1 − x^{3k−1})(1 − x^{3k})(1 − x^{3k+1})^{2}(1 − x^{3k+2})}

= \frac{(1 − x)}{(1 − x)^{2}(1 − x^{2})} ∏_{k>0} \frac{1}{(1 − x^{3k+1})(1 − x^{3k+2})}

= ∏_{k>0} \frac{1 − x^{3k}}{1 − x^{k}}

We now define the set of Kleshchev multipartitions; in order to do this we need some notation. A partition of an integer m is a sequence λ = (λ_{1}, λ_{2}, …) of non–negative integers such that λ_{1} ≥ λ_{2} ≥ ⋯ and ∑_{i} λ_{i} = m; we write |λ| = m. A multipartition of n (with r–components), is an ordered r–tuple λ = (λ^{(1)}, …, λ^{(r)}) of partitions such that |λ| = |λ^{(1)}| + ⋯ + |λ^{(r)}| = n. Let Π_{n} be the set of all multipartitions of n.

The diagram of λ ∈ Π_{n} is the set

[λ] = { (a, b, c) | 1 ≤ c ≤ r and 1 ≤ b ≤ λ_{a}^{(c)} } .

The elements of [λ] are the nodes of λ; more generally, a node is any element of ℕ × ℕ × ℕ. If γ = (a, b, c) and γ′ = (a′, b′, c′) are two nodes then γ is below γ′, or γ′ is above γ, if either c > c′, or c = c′ and a > a′. If γ = (a, b, c) ∈ [λ] is a node then its residue, relative to {q, Q_{1}, …, Q_{r}}, is res(γ) = q^{b−a}Q_{c}, an element of R. If res(γ) = x then we say that γ is an x–node.

Write λ → μ if μ is a multipartition of n + 1 such that [λ] ⊂ [μ]. In this case [μ] = [λ] ∪ {γ} for some node γ. We say that γ is an addable node of λ and a removable node of μ. If res(γ) = x then we shall also write λ →^{x} μ.

Let μ be a multipartition and suppose that η is a removable node of μ with res(η) = x. Then η is a normal x–node if whenever γ is an addable x–node in μ which is below η then there are more removable x–nodes between γ and η than there are addable x–nodes. In addition, η is good if it is the highest normal x–node in μ.

If η is a good x–node of μ and λ is the multipartition such that [μ] = [λ]∪{η} we write λ ↠^{x} μ. Notice that if λ ↠^{x} μ then λ →^{x} μ.

Definition Suppose that n ≥ 0 and that Q_{1}, …, Q_{r} are all non–zero. The set 𝒦_{n} = 𝒦_{n}(q, Q_{1} …, Q_{r}) of Kleshchev multipartitions is defined inductively as follows.

(i) 𝒦_{0} = {((0), …, (0))}; and

(ii) 𝒦_{n+1} = { μ ∈ Π_{n+1} | λ ↠^{x} μ for some λ ∈ 𝒦_{n} and some x ∈ R }.

There exist multipartitions which have no good or normal nodes so it is not completely obvious that 𝒦_{n} ≠ ∅. However, ((0), …, (0), (1^{n})) ∈ 𝒦_{n} for all n, so 𝒦_{n} is always non–empty.

Example Suppose that q = ^{3}\sqrt{1}, Q_{1} = Q_{2} = q and Q_{3} = 1. Then ℓ = 3 and res(γ) ∈ { q^{i} | 0 ≤ i < ℓ } for all nodes γ. Let λ = ((5, 4, 2), (6, 3, 2, 1), (6, 5, 3))); although one can use the definition directly to find the good nodes in λ the following procedure is easier. In the diagram below we record the residues in λ by writing i if the residue is q^{i}.

( \begin{matrix}1 & 2 & 0 & 1 & 2 \\ 0 & 1 & 2 & 0 ◯ \\ 2 & 0 \end{matrix} ,  1 2 0 1 2 0 0 1 2 2 0 1 ,  \begin{matrix}0 & 1 & 2 & 0 & 1 & 2 \\ 2 & 0 & 1 & 2 & 0 ◯ \\ 1 & 2 & 0 \end{matrix} )

We find the good node in λ of residue q^{0} = 1. Reading nodes from the bottom up, we obtain the sequence

AR R AARARR R A

where each “A” corresponds to an addable q^{0}–node and each “R” to a removable q^{0}–node. From this sequence, remove all occurrences of the string “AR”, and keep on doing this until all such strings have been deleted. The “R”s that remain are the normal q^{0}–nodes of λ and the highest of these is the good q^{0}–node. In this example, all of the grouped letters disappear; so the normal nodes are the circled nodes in the diagram and the good q^{0}–node is (2, 4, 1). Notice that λ is not a Kleshchev multipartition because the node (3, 3, 3) can never be good.

When r = 1, Π_{n} can be identified with the set of partitions of n and it is not difficult to see that 𝒦_{n} is the set of ℓ–restricted partitions. The definition of good and normal nodes in this case is due to Kleshchev [19] who used them to describe the socle of the restriction of the irreducible modules for the symmetric group in characteristic ℓ. When r > 1 we do not know of a simple description of the set 𝒦_{n} (unless q = −1; see [22]).

1 Specht modules and Grothendieck groups

Let ℋ_{R,n} be a cyclotomic Hecke algebra, as defined in the introduction. When R = ℚ(q, Q_{1}, …, Q_{r}) the simple ℋ_{R,n} –modules were constructed by the first author and Koike in [3]; for arbitrary fields the set of simple modules was first constructed by Graham and Lehrer [13], as we now describe (see also [8]).

Given two multipartitions λ = (λ^{(1)}, …, λ^{(r)}) and μ = (μ^{(1)}, …, μ^{(r)}) say that λ dominates μ, and write λ ⊵ μ, if for a = 1, 2 …, r and for all i ≥ 1

∑_{b=1}^{a−1} |λ^{(b)}| + ∑_{j=1}^{i} λ_{j}^{(a)} ≥ ∑_{b=1}^{a−1} |μ^{(b)}| + ∑_{j=1}^{i} μ_{j}^{(a)}.

If λ ⊵ μ and λ ≠ μ we write λ ▷ μ. This defines a partial order on the set of multipartitions.

Graham and Lehrer showed that for any ring R and any multipartition λ of n there exists a right ℋ_{R,n} –module S_{R}^{λ}, called a Specht module (or cell module). When R = ℚ(q, Q_{1}, …, Q_{r}), the Specht modules are simple ℋ_{R,n} –modules and coincide with the simple modules defined in [3]. In general, S_{R}^{λ} possesses a natural bilinear form and the radical, rad S_{R}^{λ}, of this form is an ℋ_{R,n} –submodule of S_{R}^{λ}. Let R be a field and D_{R}^{λ} = S_{R}^{λ}/ rad S_{R}^{λ}; then D_{R}^{λ} is either absolutely irreducible or 0.

Thus, given partitions λ and μ of n with D_{R}^{μ} ≠ 0 we can talk of the composition multiplicity d_{λμ} of D_{R}^{μ} in S_{R}^{λ}. Graham and Lehrer proved the following fundamental result.

1.1 (Graham–Lehrer [13]) Suppose that R is a field.

(i) Then { D_{R}^{λ} | λ ∈ Π_{n} and D_{R}^{λ} ≠ 0 } is a complete set of non–isomorphic irreducible ℋ_{R,n} –modules. Moreover, whenever D_{R}^{λ} is non–zero it is absolutely irreducible.

(ii) Let λ and μ be multipartitions of n and suppose that D_{R}^{μ} ≠ 0. Then d_{μμ} = 1

and d_{λμ} ≠ 0 only if λ ⊵ μ. (iii) The Hecke algebra ℋ_{R,n} is semisimple if and only if S_{R}^{λ} = D_{R}^{λ} for all

multipartitions λ ∈ Π_{n}.

For i = 1, 2, …, n let L_{i} = q^{1−i}T_{i−1} … T_{1}T_{0}T_{1}, … T_{i−1} and set c_{n} = L_{1} + L_{2} +⋯+L_{n}. Because c_{n} is a symmetric polynomial in the elements L_{1}, …, L_{n} it follows from [3, Lemma 3.3] that c_{n} belongs to the centre of ℋ_{R,n} . The next result describes how c_{n} acts upon the Specht modules.

1.2 Lemma Suppose that R is a field and let λ be a multipartition of n. Then c_{n} is central in ℋ_{R,n} and acts on the Specht module S_{R}^{λ} as multiplication by

c_{n}(λ) := ∑_{γ∈[λ]} res(γ).

Proof. When R = ℂ(q, Q_{1}, …, Q_{r}) the Specht module S_{R}^{λ} is irreducible and c_{n} must act upon S_{R}^{λ} as multiplication by a scalar because it is central; that this scalar is c_{n}(λ) follows from [3, Prop. 3.16]. By restriction, the Lemma also holds when R = ℤ[q, Q_{1}, …, Q_{r}]; hence, the general case follows by specialization.

Fix a partition {Q_{1}, …, Q_{r}} = S_{0} ⊔ S_{1} ⊔ … ⊔ S_{a} be of the parameter set as in Theorem A and write S_{k} = {Q_{k_{1}}, …, Q_{k_{r_{k}}} } for k = 0, 1, …, a. Given a multipartition λ = (λ^{(1)}, …, λ^{(r)}) let λ_{k} = (λ^{(k_{1})}, …, λ^{(k_{r_{k}})}) and set n_{k} = |λ_{k}|. Then D_{R}(λ) = D_{R}^{λ_{0}}⊗⋯⊗D_{R}^{λ_{a}} is an ℋ_{R,n_{0}}(q; S_{0})⊗_{R}⋯⊗_{R}ℋ_{R,n_{a}}(q; S_{a})–module and, as described in (1.4)(ii) below, we can define the induced ℋ_{R,n} –module Ind D_{R}(λ).

Key to the proof of our main results is the following reduction theorem which we prove using results of Rogawski [24] and Vigneras [27]; the result is also a consequence of the main result of Dipper–Mathas [9].

1.3 Theorem Suppose that R is a field and let λ be a multipartition of n and maintain the notation above. Then D_{R}^{λ} ≅ Ind D_{R}(λ).

Before proving the theorem we need to setup some notation. Let ℒ be the abelian subalgebra of ℋ_{R,n} generated by L_{1}, …, L_{n}. Over a field, the irreducible representations of ℒ are one dimensional and are labelled by the set X of weights. An irreducible representation χ ∈ X is uniquely determined by the sequence (χ(L_{1}), …, χ(L_{n})). If χ′ ∈ X and the sequence (χ′(L_{1}), …, χ′(L_{n})) is obtained from that for χ by interchanging χ(L_{i}) and χ(L_{i+1}) then we write χ′ = s_{i}χ.

If M is an ℋ_{R,n} –module and χ ∈ X let M_{χ}^{gen} be the generalized eigenspace of M with respect to χ.

Let S_{0}, …, S_{a} be as above and denote the <q>–orbit containing S_{k} by Ŝ_{k}. We define X_{red} to be the set of weights whose values satisfy χ(L_{i}) ∈ Ŝ_{k} for i = n_{0} + ⋯ + n_{k−1} + 1, …, n_{0} + ⋯ + n_{k}. Suppose that M is an ℋ_{R,n} –module and define M_{red} = ⨁_{χ∈X_{red}} M_{χ}^{gen}. Then M_{red} is a generalized eigenspace for the polynomials in L_{1}, …, L_{m} which are symmetric with respect to 𝔖_{n_{0}} ×⋯×𝔖_{n_{a}}. Thus M_{red} is stable under L_{1}, …, L_{n} and T_{w}, for w ∈ 𝔖_{n_{0}} × ⋯ 𝔖_{n_{a}}, since these elements commute with the partially symmetric polynomials in L_{1}, …, L_{n} by [3, 3.3].

1.4 To prove Theorem 1.3 it is enough to show the following.

(i) M_{red} is an ℋ_{R,n_{0}}(q; S_{0}) ⊗_{R} ⋯ ⊗_{R} ℋ_{R,n_{a}}(q; S_{a})–module. (ii) M_{red} ⊗ ℋ_{n} ≃ M. (iii) M is irreducible if and only if M_{red} is irreducible.

Let Ĥ be the algebra generated by T, X and Y whose relations are

(T − q)(T + 1) = 0, q^{−1}T XT = Y and XY = Y X.

We do not assume that X and Y are invertible. Let I(α, β) be the two dimensional representation of Ĥ given by

T ↦ ( \begin{matrix}0 & q \\ 1 & q − 1 \end{matrix} ) X ↦ ( \begin{matrix}α & −(q − 1)β \\ 0 & β \end{matrix} ) , and Y ↦ ( \begin{matrix}β & (q − 1)β \\ 0 & α \end{matrix} ) .

It is easy to see that it is irreducible if and only if β ≠ q^{±1}α, and in this case we have I(α, β) ≃ I(β, α).

1.5 Lemma Suppose that M is an ℋ_{R,n} –module and view M_{χ}^{gen} + M_{s_{i}χ}^{gen} as an Ĥ–module via

T ↦ T_{i}, X ↦ L_{i} and Y ↦ L_{i+1}.

Assume that χ(L_{i}) ≠ q^{±1}χ(L_{i+1}). Then the following hold.

(i) The composition factors of M_{χ}^{gen} +M_{s_{i}χ}^{gen} are of the form I(χ(L_{i}), χ(L_{i+1})). (ii) dimM_{χ}^{gen} = dimM_{s_{i}χ}^{gen}.

(iii) Suppose that N ≥ 0. Then (L_{i+1} − Q)^{N} acts as 0 on M_{χ}^{gen} if and only if (L_{i} − Q)^{N} acts as 0 on M_{s_{i}χ}^{gen}.

Proof. (i) Consider the radical series of M_{χ}^{gen} + M_{s_{i}χ}^{gen}. Then each successive subquotient is a semisimple module and each the simple factor contains a simultaneous eigenvector u of X and Y. As a multiset, the eigenvalues of X and Y are {χ(L_{i}), χ(L_{i+1})}. Therefore, u and T u span the simple factor, and this factor is isomorphic to one of I(χ(L_{i}), χ(L_{i+1})) or I(χ(L_{i+1}), χ(L_{i})).

(ii) Set α = χ(L_{i}) and β = χ(L_{i+1}). If α = β there is nothing to prove; hence we may assume that α ≠ β. Notice that the vectors (1, 0)^{T} and ((q−1)β, α−β)^{T} are simultaneous eigenvectors in I(α, β) of X and Y, and that the eigenvalues of X and Y on one vector are obtained by interchanging the eigenvalues of the other. Since M_{χ}^{gen}+M_{s_{i}χ}^{gen} is a direct sum of uniserial modules whose composition factors have the form I(α, β), this implies (ii).

(iii) As in (ii), we consider the direct summands of M_{χ}^{gen} + M_{s_{i}χ}^{gen}; these are uniserial modules. Now, (L_{i+1} − Q)^{N} acts as 0 on M_{χ}^{gen} if and only if Q = χ(L_{i+1}) and the lengths of the direct summands of M_{χ}^{gen} less than or equal to N. Similarly, (L_{i} − Q)^{N} acting as 0 on M_{s_{i}χ}^{gen} is equivalent to this same condition.

We are now in a position to prove conditions (i)–(iii) from (1.4). It is easy to see that M_{red} is a module for the affine Hecke algebra Ĥ_{n_{0}} ⊗_{R} ⋯ ⊗_{R} Ĥ_{n_{a}}. We check the conditions for L_{i} , for i = 1, n_{0} + 1, . . .. If χ ∈ X_{red} and i = n_{0}+⋯+n_{k−1}+1 then χ(L_{i}) must be an element in S_{k}, since χ(L_{i}) must appear as an eigenvalue of L_{1} by Lemma 1.5(ii). Let Q be an element in S_{k} and denote its multiplicity in S_{k} by n_{Q}. Then Lemma 1.5(iii) says that (L_{i} − Q)^{n_{Q}} = 0 on M_{χ}^{gen} for those χ ∈ X_{red} which satisfy χ(L_{i}) = Q. Taking the product of these factors (L_{i} − Q)^{n_{Q}}, we see that M_{red} is an H_{R,n_{0}}(q; S_{0}) ⊗_{R} ⋯ ⊗_{R} H_{R,n_{a}}(q; S_{a})– module. We have proved (1.4)(i).

By the argument of [24, Proposition 4.1], as an Ĥ_{n}–module, M is isomorphic to the induced module of M_{red}. Since the spaces Ĥ_{n}/Ĥ_{n_{0}} ⊗ ⋯ ⊗ Ĥ_{n_{a}} and H_{n}/H_{n_{0}} ⊗ ⋯ ⊗ H_{n_{a}} have the same dimension, this gives (1.4)(ii).

Finally, (1.4)(iii) follows because the functors M ↦ M_{red} and N ↦ ℋ_{R,n} ⊗N are both exact.

This completes the proof of Theorem 1.3.

Thus, because of Theorem 1.3, in order to prove our main results we can reduce to the cases where the parameters Q_{1}, …, Q_{r} are either (i) all zero, or (ii) all units in R and Q_{i}/Q_{j} is a power of q for all 1 ≤ i, j ≤ r. We next dispense with case (i) by classifying the irreducible ℋ_{R,n} –modules when Q_{1} = ⋯ = Q_{r} = 0.

Recall that ℓ is the smallest positive integer such that 1 + q + ⋯ + q^{ℓ−1} = 0; if no such integer exists then ℓ = ∞. A partition λ = (λ_{1}, λ_{2}, …) is ℓ–restricted if λ_{i} − λ_{i+1} < ℓ for all i ≥ 1. In particular, all partitions are ∞–restricted. Let Λ_{0}^{+} be the set of all multipartitions λ = (λ^{(1)}, …, λ^{(r)}) of n such that λ^{(r)} is ℓ–restricted and λ^{(j)} = (0) for 1 ≤ j < r.

1.6 Theorem Suppose that R is a field and that Q_{1} = ⋯ = Q_{r} = 0.

(i) Let λ = (λ^{(1)}, …, λ^{(r)}) be a multipartition of n. Then D_{R}^{λ} is non–zero if and only if λ ∈ Λ_{0}^{+}.

(ii) { D_{R}^{λ} | λ ∈ Λ_{0}^{+} } is a complete set of non–isomorphic irreducible right ℋ_{R,n} – modules.

Before we can establish the theorem we have to recall some notation from [8] and prove a preliminary lemma.

In [8, Theorem 3.26] it is shown that ℋ_{R,n} is a cellular algebra with a cellular basis {m_{𝔰𝔱}}, where 𝔰 and 𝔱 run over all pairs of standard λ–tableaux of the same shape for all multipartitions of n. The dominance order ⊵ extends naturally to the set of standard tableaux and for each λ there is a unique tableau 𝔱^{λ} such that 𝔱^{λ} ⊵ 𝔰 for all λ–tableau 𝔰. These details can be found in [8].

Given a multipartition λ let N̄^{λ} be the submodule of ℋ_{R,n} spanned by the basis elements {m_{𝔲𝔳}}, where 𝔲 and 𝔳 are standard μ–tableaux for some multipartition μ such that μ ▷ λ. Then N̄^{λ} is a two–sided ideal of ℋ_{R,n} and the Specht module S_{R}^{λ} is isomorphic to the R–module with basis m_{𝔰} = N̄^{λ} + m_{𝔱^{λ}𝔰}, where 𝔰 runs over the set of standard λ–tableaux. Moreover, there is a natural bilinear form ⟨ , ⟩ defined on S_{R}^{λ} which is determined by

⟨m_{𝔰}, m_{𝔱}⟩m_{𝔱^{λ}𝔱^{λ}} ≡ m_{𝔱^{λ}𝔰}m_{𝔱𝔱^{λ}} mod N̄^{λ}.

Finally, D_{R}^{λ} = S_{R}^{λ}/ rad S_{R}^{λ}, where rad S_{R}^{λ} = { x ∈ S_{R}^{λ} | ⟨x, y⟩ = 0 for all y ∈ S_{R}^{λ} }.

1.7 Lemma Suppose that Q_{1} = ⋯ = Q_{r} = 0 and that λ is a multipartition of n. Let k be an integer with 1 ≤ k ≤ n. Then m_{𝔱^{λ}𝔰}L_{k} ∈ N̄^{λ} for any standard λ–tableau 𝔰.

Proof. We argue by induction on 𝔰. When 𝔰 = 𝔱^{λ} the result is a special case of [15, Prop. 3.7]. If 𝔰 ≠ 𝔱^{λ} then there exists an integer i such that 𝔱 = 𝔰(i, i + 1) ▷ 𝔰 and 1 ≤ i < n. Therefore, m_{𝔱^{λ}𝔰}L_{k} = m_{𝔱^{λ}𝔱}T_{i}L_{k}. If i ≠ k − 1 and i ≠ k then T_{i} and L_{k} commute so the lemma follows by induction. The remaining cases also follow by induction on 𝔰 because T_{k−1}L_{k} = (q−1)L_{k}+L_{k−1} and T_{k}L_{k} = qL_{k+1}T_{k}^{−1}.

Proof of Theorem 1.6. Because ℋ_{R,n} is a cellular algebra, we need only consider part (i). For this, recall that D_{R}^{λ} ≠ 0 if and only if rad S_{R}^{λ} ≠ S_{R}^{λ}. If 𝔰 and 𝔱 are standard λ–tableaux then the bilinear form on S_{R}^{λ} is determined by

⟨m_{𝔰}, m_{𝔱}⟩m_{𝔱^{λ}𝔱^{λ}} ≡ m_{𝔱^{λ}𝔰}m_{𝔱𝔱^{λ}} mod N̄^{λ}.

First suppose that λ ∈ Λ_{0}^{+}. Then for all standard λ–tableaux 𝔰 and 𝔱 we have that m_{𝔱^{λ}𝔰}m_{𝔱𝔱^{λ}} ∈ ℋ_{R,n} (𝔖_{n}) and so it follows from [7, Theorem 6.3] that D_{R}^{λ} ≠ 0. (Note that in the notation of [7] we are working with dual Specht modules and so partitions must be conjugated.)

To prove the converse, we claim that if λ ∉ Λ_{0}^{+} then ⟨m_{𝔰}, m_{𝔱}⟩ = 0 for all standard λ–tableaux 𝔰 and 𝔱; in particular, this will show that rad S_{R}^{λ} = S_{R}^{λ} and so complete the proof of the theorem. As in [8, Defn. 3.5], write m_{𝔱^{λ}𝔱^{λ}} = m_{λ} = u_{a}^{+}x_{λ}; all that we need to know about this factorization is that L_{1} is a factor of u_{a}^{+} if λ ∉ Λ_{0}^{+}. By the above remarks,

⟨m_{𝔰}, m_{𝔱}⟩m_{𝔱^{λ}𝔱^{λ}} ≡ m_{𝔱^{λ}𝔰}m_{𝔱𝔱^{λ}} = m_{𝔱^{λ}𝔰}T_{d(𝔱)}^{∗}x_{λ}u_{a}^{+} ≡ ∑_{𝔰′} a_{𝔰′}m_{𝔱^{λ}𝔰′}u_{a}^{+} mod N̄^{λ},

for some a_{𝔰′} ∈ R. So in order to prove our claim it suffices to show that m_{𝔱^{λ}𝔰′}u_{a}^{+} ∈ N̄^{λ} for all standard λ–tableaux 𝔰′. However, we have already noted that L_{1} is a factor of u_{a}^{+} when λ ∉ Λ_{0}^{+}; so, Lemma 1.7 proves the claim and hence the theorem.

Thus, in order to prove Theorems A, C and D we are reduced to the following situation.

1.8 Henceforth, we assume that R is a field, q ≠ 1 and that there exist integers n_{i} such that Q_{i} = q^{n_{i}} for i = 1, 2, …, r. Let ℓ be the smallest positive integer such that q^{ℓ} = 1; set ℓ = ∞ if no such integer exists. If ℓ is finite let I_{ℓ} = {0, 1, …, ℓ − 1}; otherwise, let I_{ℓ} = ℤ.

We also write λ ⟶^{i} μ and λ ↠^{i} μ rather than λ →^{q^{i}} μ and λ ↠^{q^{i}} μ respectively.

Let 𝔲_{R,n} = 𝒢_{0}(ℋ_{R,n} ) ⊗_{ℤ} ℂ be the Grothendieck group of finitely generated ℋ_{R,n} –modules modulo short exact sequences, with coefficients extended to ℂ; see [6, §16]. Given a finitely generated ℋ_{R,n} –module let [M] denote its equivalence class in 𝔲_{R,n}. By (1.1), 𝔲_{R,n} is an abelian group which is free as a ℂ–module with basis { [D_{R}^{μ}] | μ ∈ Π_{n} and D_{R}^{μ} ≠ 0 }. Furthermore, [S_{R}^{λ}] = ∑_{λ} d_{λμ}[D_{R}^{μ}] if μ ∈ Π_{n} and D_{R}^{μ} ≠ 0.

Given any ℋ_{R,n} –module M let Res M be the restriction of M to ℋ_{R,n−1} . Then Res is an exact functor from the category of right ℋ_{R,n} –modules to the category of right ℋ_{R,n−1} –modules. Similarly, we have an adjoint induction functor Ind given by Ind M = M ⊗_{ℋ_{R,n}} ℋ_{R,n+1} . These functors induce maps between the corresponding Grothendieck groups via Res [M] = [Res M] and Ind [M] = [Ind M].

1.9 Proposition Suppose that λ is a multipartition of n. Then Res S_{R}^{λ} has a filtration with composition factors precisely the Specht modules S_{R}^{ν} where ν is a multipartition of n − 1 and ν → λ.

Proof. As above, let {m_{𝔱}} be the standard basis of the Specht module S_{R}^{λ}. If 𝔱 is a standard λ–tableau let 𝔱 ↓ n − 1 be the tableau obtained by deleting the node labelled n and let |𝔱 ↓ n − 1| be the corresponding multipartition. By repeating the proof of [8, 3.15,3.18], if h ∈ ℋ_{R,n−1} then m_{𝔱}h is a linear combination of terms m_{𝔳} such that |𝔳 ↓ n − 1| ⊵ |𝔱 ↓ n − 1|; that is, n occupies either the same node or a lower node in 𝔳 than it does in 𝔱. Therefore, by extending the dominance order on the set { ν | ν → λ } to a total order, we can define an ℋ_{R,n−1} –stable filtration of S_{R}^{λ} where for each quotient there exists a multipartition ν such that ν → λ and the quotient has a basis indexed by the λ–tableau 𝔱 such that |𝔱 ↓ n − 1| = ν. As in [8, 3.20], for each such ν, we can define a map from S_{R}^{ν} into the corresponding quotient of the filtration which maps m_{𝔰} to the image of m_{𝔱} in the quotient where 𝔰 = 𝔱 ↓ n − 1. From what we have said, this is an ℋ_{R,n−1} –module homomorphism; by a counting argument, it is actually an isomorphism.

1.10 Corollary Let λ be a multipartition of n. Then

Res [S_{R}^{λ}] = ∑_{ν→λ}[S_{R}^{ν}] and Ind [S_{R}^{λ}] = ∑_{λ→μ}[S_{R}^{μ}].

Proof. The restriction formula is a direct consequence of Proposition 1.9. The induction formula follows by adjointness.

By (1.1), [D_{R}^{μ}] is a linear combination of Specht modules [S_{R}^{λ}] so this completely determines the maps Res and Ind at the level of Grothendieck groups.

Now consider the element c_{n} and suppose that D_{R}^{μ} ≠ 0. By Lemma 1.2, c_{n} is central and acts on S_{R}^{μ}, and hence also on D_{R}^{μ}, as multiplication by c_{n}(μ). Let M be a ℋ_{R,n} –module and let M = ⊕_{x}M_{x} be the decomposition of M into a direct sum of generalized eigenspaces M_{x} for the action of c_{n}. Then, given i ∈ I_{ℓ} we have functors i–Res and i–Ind given by

(1.11) i–ResM = ⨁_{x∈R}(Res M_{x})_{x−q^{i}} and i–IndM = ⨁_{x∈R}(Ind M_{x})_{x+q^{i}}.

Then Res = ∑_{i} i–Res and Ind = ∑_{i} i–Ind. In turn, these functors induce homomorphisms i–Res : 𝔲_{R,n} ⟶ 𝔲_{R,n−1} and i–Ind : 𝔲_{R,n} ⟶ 𝔲_{R,n+1} given by i–Res [M] = [i–ResM] and i–Ind [M] = [i–IndM]. Comparing the definitions and Corollary 1.10 we can rephrase this as follows.

1.12 Corollary Suppose that λ is a multipartition of n and let i ∈ I_{ℓ}. Then the homomorphisms i–Res : 𝔲_{R,n} ⟶𝔲_{R,n−1} and i–Ind : 𝔲_{R,n} ⟶𝔲_{R,n+1} are completely determined by

i–Res [S_{R}^{λ}] = ∑_{ν⟶^{i}λ}[S_{R}^{ν}] and i–Ind [S_{R}^{λ}] = ∑_{λ⟶^{i}μ}[S_{R}^{μ}].

Although we are most interested 𝔲_{R,n} it is more useful to consider 𝔲_{R,n}^{0}, the Grothendieck group of finitely generated projective modules, again with coefficients extended to ℂ. If P is a projective ℋ_{R,n} –module write [[P]] for its image in 𝔲_{R,n}^{0}. For each multipartition μ with D_{R}^{μ} ≠ 0, up to isomorphism, there is a uniquely determined projective indecomposable module P_{R}^{μ} and 𝔲_{R,n}^{0} is the free abelian group which is free as a ℂ–module with basis { [[P_{R}^{μ}]] | μ ∈ Π_{n} and D_{R}^{μ} ≠ 0 }.

It is easy to see that the functors Res and Ind take projectives to projectives and so again induce maps between 𝔲_{R,n}^{0} and 𝔲_{R,n±1}^{0}. Moreover, by Lemma 1.2, each projective decomposes into a sum of generalized eigenspaces of c_{n}, which are again projective modules, so we also have homomorphisms i–Res and i–Ind which are defined using (1.11) exactly as before.

The next Lemma is standard.

1.13 Lemma Let 𝔲_{R,n}^{∗} = Hom_{ℂ}(𝔲_{R,n}, ℂ).

(i) The abelian groups 𝔲_{R,n}^{0} and 𝔲_{R,n}^{∗} are canonically isomorphic. (ii) The natural pairing 𝔲_{R,n}^{0} × 𝔲_{R,n} → ℂ is determined by ([[P_{R}^{λ}]], [D_{R}^{μ}]) = δ_{λμ}. (iii) If P is a projective ℋ_{R,n} –module and M is any ℋ_{R,n+1} –module then

(i–Ind [[P]], [M]) = ([[P]], i–Res [M]).

(iv) If P is a projective ℋ_{R,n} –module and M is any ℋ_{R,n−1} –module then

(i–Res [[P]], [M]) = ([[P]], i–Ind [M]).

Proof. (i) Given any projective ℋ_{R,n} –module P the functor Hom_{ℋ_{R,n}} (P, ) is exact; hence, we can define a map d_{P} ∈ 𝔲_{R,n}^{∗} by d_{P} [M] = dim Hom_{ℋ_{R,n}} (P, M). It follows that P ↦ d_{P} is an isomorphism of abelian groups.

(ii) The natural pairing between 𝔲_{R,n}^{∗} and 𝔲_{R,n} is given by evaluation; therefore, using the notation of (i),

([[P_{R}^{λ}]], [D_{R}^{μ}]) = d_{P_{R}^{λ}}[D_{R}^{μ}] = dim Hom_{ℋ_{R,n}} (P_{R}^{λ}, D_{R}^{μ}) = δ_{λμ}.

(iii) Using the adjointness of i–Res and i–Ind we find that

(i–Ind [[P]], [M]) = dim Hom_{ℋ_{R,n}} (i–IndP, M) = dim Hom_{ℋ_{R,n}} (P, i–ResM)
= ([[P]], i–Res [M]).

The proof of (iv) is similar.

Henceforth, we identify 𝔲_{R,n}^{0} and 𝔲_{R,n}^{∗}.

There is a natural homomorphism c : 𝔲_{R,n}^{0} ⟶𝔲_{R,n} given by [[P]] ↦ [P]. The map c is the so–called “Cartan map” and it is best understood as follows. Let ℱ_{n} be the free abelian group with ℂ–basis the set of symbols { [[S^{λ}]] | λ ∈ Π_{n} }. We define group homomorphisms Res, i–Res, Ind and i–Ind from ℱ_{n} to ℱ_{n±1} by formulas in Corollary 1.12; importantly, these maps are independent of the field R.

By [13, Theorem 3.7], the projective indecomposable P_{R}^{μ} has a filtration with each composition factor isomorphic to some Specht module such that S_{R}^{λ} occurs with multiplicity d_{λμ}. Since the [[P_{R}^{μ}]] are linearly independent, we have a well–defined injective homomorphism of abelian groups e : 𝔲_{R,n}^{0} ⟶ ℱ_{n} given by e[[P_{R}^{μ}]] = ∑_{λ} d_{λμ}[[S^{λ}]]. Similarly, there is a well–defined surjective homomorphism d : ℱ_{n} ⟶ 𝔲_{R,n} given by d[[S^{λ}]] = [S_{R}^{λ}] = ∑_{μ} d_{λμ}[D_{R}^{μ}]. Moreover, by [13, Theorem 3.7] the diagram

[figure: d; ℱ_{n}; e; 𝔲_{R,n}^{0}; 𝔲_{R,n}; c]

commutes. By (1.1)(iii), ℱ_{n} is isomorphic to the Grothendieck group of a semisimple Hecke algebra, so this is really just the Cartan–Brauer cde–triangle for ℋ_{R,n} (see [6, §18]).

Finally, it follows directly from the definitions that all of the maps in the Cartan triangle commute with the homomorphisms i–Res and i–Ind. We summarize these results in the following Lemma.

1.14 Lemma Suppose that R is a field and that i ∈ I_{ℓ}. Then all of the squares and triangles in the diagram

[figure: d; ℱ_{n−1}; e; 𝔲_{R,n−1}^{0}; 𝔲_{R,n−1}; c; d; ℱ_{n}; e; 𝔲_{R,n}^{0}; 𝔲_{R,n}; d; ℱ_{n+1}; e; 𝔲_{R,n+1}^{0}; 𝔲_{R,n+1}; i–Res; i–Ind; i–Res; i–Ind; i–Res; i–Ind]

commute (the vertical arrows are the Cartan maps c).

2 The Kac–Moody algebra

Throughout this section we maintain the assumptions of (1.8). In particular, ℓ is the smallest positive integer such that q^{ℓ} = 1; ℓ = ∞ if no such integer exists.

If ℓ is finite let U(𝔤) be the Kac–Moody algebra of type A_{ℓ−1}^{(1)}; if ℓ = ∞ let U(𝔤) be the Kac–Moody algebra of type A_{∞}. We refer the reader to Kac’s book [17] for the standard properties of Kac–Moody algebras.

The Kac–Moody algebra U(𝔤) is generated by elements e_{i}, f_{i}, h_{i} and d where i ∈ I_{ℓ}. Let U(𝔥) be the Cartan subalgebra of U(𝔤), the subalgebra of U(𝔤) generated by h_{i} and d, and for i ∈ I_{ℓ} let Λ_{i}, α_{i} ∈ 𝔥^{∗} be the fundamental and simple weights of U(𝔤). We may choose d so that Λ_{i}(d) = 0 and α_{i}(d) = δ_{i0} for all i ∈ I_{ℓ}.

The following result is a theorem of Hayashi [14] when r = 1 (see also [23]); the general case follows easily; see [1, 4.5]. We give the proof of a slightly stronger result in Proposition 2.6 below.

2.1 Definition Given i ∈ I_{ℓ} let N_{i}(λ) = # { μ | λ ⟶^{i} μ } − # { ν | ν ⟶^{i} λ } and let N_{d}(λ) = # { (a, b, c) ∈ [λ] | res(a, b, c) = 0 }.

2.2 Lemma Let ℱ = ⨁_{n≥0} ℱ_{n}. Then ℱ becomes a U(𝔤)–module with action

e_{i}[[S^{λ}]] = ∑_{ν⟶^{i}λ}[[S^{ν}]], f_{i}[[S^{λ}]] = ∑_{λ⟶^{i}μ}[[S^{μ}]], h_{i}[[S^{λ}]] = N_{i}(λ)[[S^{λ}]],

for each i ∈ I_{ℓ}, and d[[S^{λ}]] = −N_{d}(λ)[[S^{λ}]].

Comparing Lemma 2.2 with Corollary 1.12, we can identify i–Ind and i–Res with f_{i} and e_{i} on ⨁ 𝔲_{R,n}^{0}.

For j = 1, 2, …, r recall that Q_{j} = q^{n_{j}} for some integer n_{j}. When ℓ is finite q^{ℓ} = 1 we may assume that 0 ≤ n_{j} < ℓ for all j. In this way we associate the dominant weight Λ = Λ_{n_{1}} + ⋯ + Λ_{n_{r}} with ℋ_{R,n} . Notice that the multiplicity of Λ_{i} in Λ is equal to the number of Q_{j} such that Q_{j} = q^{i}.

2.3 Theorem Let R be a field and set 𝔲_{R}^{0} = ⨁_{n≥0} 𝔲_{R,n}^{0}. Then 𝔲_{R}^{0} is isomorphic to the integrable highest weight U(𝔤)–module of highest weight Λ.

Proof. When R = ℂ this is just [1, Theorem 4.4(i)]: the crucial point in the proof is to show that 𝔲_{R}^{0} is a cyclic U(𝔤)–module, which was proved by counting orbits in quiver spaces which are known to parametrize both the irreducible representations of the affine Hecke algebras of type A and also a basis of the quantum algebras of affine type A. Let R be a field of characteristic 0 and let R′ be an extension of R. By (1.1)(i), D_{R}^{μ} is absolutely irreducible so D_{R′}^{μ} ≅ D_{R}^{μ} ⊗_{R} R′ and i–Ind and i–Res act in the same way upon [D_{R}^{μ}] and [D_{R′}^{μ}] for all μ. Thus, we may identify 𝔲_{R}^{0} and 𝔲_{R′}^{0} as U(𝔤)–modules. In conclusion, the Theorem holds for fields of characteristic 0 by embedding them into a suitable extension of ℂ.

Now suppose that R is a field of positive characteristic. By construction, 𝔲_{R}^{0} is a ℂ–submodule of ℱ. However, by Lemma 1.14 and Lemma 2.2, if D_{R}^{μ} is non– zero then both f_{i}[[P_{R}^{μ}]] and e_{i}[[P_{R}^{μ}]] are elements of 𝔲_{R}^{0}; so, 𝔲_{R}^{0} is actually a U(𝔤)– submodule of ℱ. Notice also that ℱ is integrable since each multipartition has a finite number of addable and removable nodes; hence, 𝔲_{R}^{0} is also integrable.

Let u_{Λ} = [[S^{((0),... ,(0))}]] be the vector in ℱ corresponding to the empty multipartition; then U(𝔤)𝔲_{Λ} ⊆ 𝔲_{R}^{0}. Now u_{Λ} is annihilated by e_{i} and by d and h_{i}u_{Λ} = Λ(h_{i})u_{Λ} for all i ∈ I_{ℓ}; thus, U(𝔤)𝔲_{Λ} is an integrable highest weight module of highest weight Λ and to complete the proof we must show that 𝔲_{R}^{0} = U(𝔤)u_{Λ}. To do this, take a modular system (K, O, R) where K is a field of characteristic 0 and q′ is an element of order ℓ in O ⊂ K which is in the preimage of q. As in section 2, we have operators i–Res and i–Ind for 𝔲_{K}^{0} and 𝔲_{R}^{0} respectively; by Corollary 1.12 these are defined by their action on Specht modules and by Corollary 1.10 they are both restrictions of the same operators acting on ℱ. By lifting idempotents, 𝔲_{R}^{0} is a submodule of 𝔲_{K}^{0}. However, we have already shown that 𝔲_{K}^{0} is the integrable highest weight module of weight Λ; so any submodule of 𝔲_{K}^{0} which contains the highest weight vector coincides with the module itself. Thus, 𝔲_{R}^{0} = U(𝔤)𝔲_{Λ} as required.

Note that Theorem 2.3 is completely independent of the field R; hence, this completes the proof of Theorem A. Proof of Theorem B. Let k be a field. We first notice that any simple Ĥ_{k,n}– module is a simple module for some ℋ_{k,n} . Hence we can use Theorem 1.3 to reduce to the case where the support of \underline{λ} is a single <q>–orbit in k. In this case, the simple Ĥ_{ℂ,n}–modules are indexed by aperiodic multisegments; see [5, Theorem 8.6.12] and [1, Proposition 4.3]. Therefore, by an similar argument to that of the first paragraph of the proof of Theorem 2.3, we deduce that the simple Ĥ_{k,n}–modules are indexed by aperiodic multisegments whenever k is a field of characteristic 0.

Now suppose that k is a field of positive characteristic and, as in [1], let 𝔘_{k}^{∗} be the graded dual of the direct sum of Grothendieck groups of Ĥ_{k,n}–modules; cf. the definition of 𝔲_{k}^{0}. Then 𝔘_{k}^{∗} is a U^{−}(𝔤)–module and 𝔲_{k}^{∗} is a quotient of 𝔘_{k}^{∗} by [1, Lemma 4.1]. By fixing n and taking the highest weight Λ sufficiently large, the degree n part of 𝔘_{k}^{∗} is isomorphic to the degree n part of 𝔲_{k}^{∗}. Since Theorem A tells us that the dimension of the degree n part of 𝔲_{k}^{∗} only depends on the multiplicative order of q, we have that the simple Ĥ_{k,n}–modules are also indexed by the aperiodic multisegments of size n. We have proved (i).

To prove (ii), take a modular system (K, O, k) where q lifts to an element of the same order in O. Recall that the isomorphism 𝔲_{K}^{0} ≃ 𝔲_{k}^{0}, which induces an isomorphism from 𝔲_{k} to 𝔲_{K} by Lemma 1.13(i), is given by the modular reduction procedure. Hence, (1.1) gives (ii).

Now we turn to the proof of Theorem C. By Theorem A, we can restrict ourselves to the case R = ℂ. The main result of [1] states that if R = ℂ then the canonical basis of 𝔲_{ℂ}^{0} coincides with the basis of 𝔲_{ℂ}^{0} given by the principal indecomposable ℋ_{ℂ,n} –modules. The canonical basis of 𝔲_{ℂ}^{0} is indexed by the vertices of the crystal graph of 𝔲_{ℂ}^{0}; to describe this we next introduce the quantized enveloping algebra of U(𝔤).

Let v be an indeterminate over ℂ and let U_{v}(𝔤) be the quantized enveloping algebra of U(𝔤). This is a ℂ(v)–algebra generated by elements E_{i}, F_{i}, K_{h_{i}} and K_{d} (for i ∈ I_{ℓ}), which are subject to the quantized Serre relations [21, 1.4.3].

We first show how ℱ_{v} = ℱ ⊗_{ℂ} ℂ(v) may be endowed with the structure of a U_{v}(𝔤)–module. For convenience, consider ℱ_{v} to be the free ℂ(v)–module with basis { λ | λ ∈ Π_{n} for some n ≥ 0 }.

2.4 Definition (cf. [20]) Let λ, ν and μ be multipartitions such that ν ⟶^{i} λ and λ ⟶^{i} μ for some i ∈ I_{ℓ}.

(i) N_{i}^{r}(ν, λ) = # { ν ⟶^{i} α | α ▷ λ } − # { β ⟶^{i} λ | ν ▷ β } .

(ii) N_{i}^{l}(λ, μ) = # { λ ⟶^{i} α | μ ▷ α } − # { β ⟶^{i} μ | β ▷ λ } .

These definitions can be rephrased in terms of the addable and removable nodes of the introduction (cf. Lemma 2.5 below).

2.5 Lemma Suppose that λ and μ are multipartitions with λ ⟶^{i} μ for some i ∈ I_{ℓ}. Let η be the addable node of λ such that [μ] = [λ] ∪ {η}. Then η is normal if and only if N_{i}^{l}(λ, μ) ≤ 0 and N_{i}^{l}(λ, μ) < N_{i}^{l}(λ, ν) whenever λ ⟶^{i} ν and μ ▷ ν.

Proof. The addable i–nodes below η are in bijection with those multipartitions α such that λ ⟶^{i} α and μ ▷ α; similarly, the removable i–nodes below η correspond to the multipartitions β with β ⟶^{i} μ and β ▷ λ. Hence, the Lemma is just a translation of the original definition given in the introduction into the notation of Definition 2.4.

2.6 Proposition The ℂ(v)–module ℱ_{v} is an integrable U_{v}(𝔤)–module with action determined by

\begin{matrix}K_{h_{i}}λ = & v^{N_{i}(λ)}λ, & K_{d}λ = & v^{−N_{d}(λ)}λ \\ E_{i}λ = & ∑_{ν⟶^{i}λ} v^{−N_{i}^{r}(ν,λ)}ν, & F_{i}λ = & ∑_{λ⟶^{i}μ} v^{N_{i}^{l}(λ,μ)}μ \end{matrix}

where i ∈ I_{ℓ} and λ is multipartition of k.

Proof. First consider the case where r = 1. In this case ℱ_{v} is the Fock space ℱ_{Q_{1}}^{(1)}; that is, the free ℂ(v)–module with basis the set of all partitions of all integers), and this action was discovered by Hayashi [14] (cf. [20]). In fact, he considered only the case where Q_{1} = 1; however, the general case is easily derived from this.

Suppose now that r > 1 and identify ℱ_{v} with ℱ_{Q_{1}}^{(1)}⊗⋯⊗ℱ_{Q_{r}}^{(1)} via the ℂ(v)– linear map which sends (λ^{(1)}, …, λ^{(r)}) to λ^{(1)} ⊗⋯⊗λ^{(r)}. Now, U_{v}(𝔤) is a Hopf algebra having a standard comultiplication map Δ; because of its compatibility with the representation theory of ℋ_{R,n} we use the twisted coproduct map Δ′ = w_{0}Δw_{0}, where w_{0} is the longest element of 𝔖_{r}. Then Δ′ is the ℂ(v)–linear map determined by

Δ′(K_{h}) = K_{h}⊗K_{h}, Δ′(E_{i}) = E_{i}⊗1+K_{−h_{i}}⊗E_{i}, and Δ′(F_{i}) = F_{i}⊗K_{h_{i}}+1⊗F_{i},

for all i ∈ I_{ℓ} and all h ∈ U(𝔥). The coproduct map Δ′ induces an action of U_{v}(𝔤) upon ℱ_{v}. For example, using the obvious notation,

E_{i}λ = E_{i}λ^{(1)} ⊗ λ^{(2)} ⊗ ⋯ ⊗ λ^{(r)} + K_{−h_{i}}λ^{(1)} ⊗ E_{i}λ^{(2)} ⊗ ⋯ ⊗ λ^{(r)}
+ ⋯ + K_{−h_{i}}λ^{(1)} ⊗ ⋯ ⊗ K_{−h_{i}}λ^{(r−1)} ⊗ E_{i}λ^{(r)}

= ( ∑_{μ^{(1)}} v^{−N_{i}^{r}(λ^{(1)},μ^{(1)})}μ^{(1)}) ⊗ λ^{(2)} ⊗ ⋯ ⊗ λ^{(r)}

+ ⋯ + v^{∑_{j=1}^{r−1} −N_{i}(λ^{(j)})}λ^{(1)} ⊗ ⋯ ⊗ λ^{(r−1)} ⊗ ( ∑_{μ^{(r)}} v^{−N_{i}^{r}(λ^{(r)},μ^{(r)})}μ)

= ∑_{λ⟶^{i}μ} v^{−N_{i}^{r}(λ,μ)}μ,

as required. The other calculations are similar.

Finally, ℱ_{v} is integrable because each basis element λ is a weight vector and E_{i}^{k}λ = F_{i}^{k}λ = 0 for all sufficiently large k since λ has only a finite number of addable and removable nodes.

By the Proposition, the empty multipartition u_{Λ} = ((0), …, (0)) in ℱ_{v} is a highest weight vector of weight Λ. Now integrable U_{v}(𝔤)–modules are completely reducible [21, 6.2.2], so the highest weight module L(Λ) := U_{v}(𝔤)·u_{Λ} is a direct summand of ℱ_{v}.

If i ∈ I_{ℓ} and k ≥ 0 let E_{i}^{(k)} = E_{i}^{k}/[k]_{v}^{!} and F_{i}^{(k)} = F_{i}^{k}/[k]_{v}^{!} where [k]_{v}^{!} = [1]_{1}[2]_{v} … [k]_{v} where [k]_{v} = (v^{k} − v^{−k})/(v − v^{−1}).

Given i ∈ I_{ℓ} let U_{i} be the subalgebra of U_{v}(𝔤) generated by E_{i}, F_{i}, and K_{±h_{i}}. Every element x of ℱ_{v} can be written in a unique way as a linear combination x = ∑_{Γ} x_{Γ} where Γ ∈ 𝔥^{∗} and K_{h}x_{Γ} = v^{Γ(h)}x for all h ∈ 𝔥. If x = x_{Γ} for some Γ ∈ 𝔥^{∗}, because E_{i} and F_{i} both act as locally nilpotent operators on ℱ, it follows from [21, 16.1.4] that

x = ∑_{s≥0} F_{i}^{(s)}x_{s},

where each x_{s} is uniquely determined by the conditions that E_{i}x_{s} = 0 and K_{i}x_{s} = v^{Γ(h_{i})+2s}x_{s}. The Kashiwara operators [18] Ẽ_{i} and F̃_{i} are defined by

(2.7)  Ẽ_{i}x = ∑_{s≥1} F_{i}^{(s−1)}x_{s} and F̃_{i}x = ∑_{s≥0} F_{i}^{(s+1)}x_{s}.

Let A be the ring of rational functions in ℂ(v) which do not have a pole at 0. Let ℱ_{A} = ⊕_{λ}Aλ where λ runs over all multipartitions of all integers. We denote the set of multipartitions by Π.

2.8 Theorem (cf. [16, Theorem 3.6]) Suppose that μ is a multipartition of n and let i ∈ I_{ℓ}.

(i) If μ has no good node of residue i then Ẽ_{i}μ = 0 (mod vℱ_{A}). (ii) If η is the good node of residue i in μ and μ = λ ∪ {η} then

F̃_{i}λ = μ (mod vℱ_{A}) and Ẽ_{i}μ = λ (mod vℱ_{A}).

(iii) (ℱ_{A}, Π) is a crystal base of ℱ_{v}.

Proof. When r = 1 this result is proved in [23]; note that they work with the crystal basis at infinity. Since (ℱ_{A}, Π) is the tensor product of the crystal bases of level 1 modules, we have (iii). To prove (i) and (ii), we recall Kashiwara’s rule for the tensor product of crystal bases.

Let (L_{1}, B_{1}), (L_{2}, B_{2}) be crystal bases and let l_{i}^{+}(b) = max { k | Ẽ_{i}^{k}b ≠ 0 } and l_{i}^{−}(b) = max { k | F̃_{i}^{k}b ≠ 0 }. Then we have

F̃_{i}(b_{1} ⊗ b_{2}) = \{ \begin{matrix}F̃_{i}(b_{1}) ⊗ b_{2}, & if l_{i}^{+}(b_{1}) ≥ l_{i}^{−}(b_{2}), \\ b_{1} ⊗ F̃_{i}(b_{2}), & if l_{i}^{+}(b_{1}) < l_{i}^{−}(b_{2}). \end{matrix}

Note that we adopt Δ′ as coproduct. Assume that we have already proved (i) and (ii) for r and consider λ ⊗ λ^{(r+1)} ∈ ℱ_{v} ⊗ ℱ_{Q_{r+1}}^{(1)}. Recall the procedure described in the introduction for deleting AR’s from the sequence of addable and removable nodes for this multipartition. In the final sequence, the number of R’s remaining which came from λ is l_{i}^{+}(λ), and the number of A’s remaining from λ^{(r+1)} is l_{i}^{−}(λ^{(r+1)}). Thus, if l_{i}^{+}(λ) ≥ l_{i}^{−}(λ^{(r+1)}), all of the A’s in λ^{(r+1)} are cancelled and F̃_{i}(λ ⊗ λ^{(r+1)}) is obtained by changing the lowest A in λ into an R. If l_{i}^{+}(λ) > l_{i}^{−}(λ^{(r+1)}) then some of the A’s from λ^{(r+1)} still remain and the bottom A is changed into an R. This rule is exactly the one we claimed for F̃_{i}. The proof for Ẽ_{i} is similar.

Let L(Λ)_{A} = F_{A} ∩ L(Λ) and B_{0}(Λ) = { F̃_{i_{1}} … F̃_{i_{k}}u_{Λ} + vL(Λ)_{A} | i_{j} ∈ I_{ℓ} } \ {0}. Then (L(λ)_{A}, B_{0}(Λ)) is a crystal base. We can now describe the crystal graph of L(Λ).

2.9 Definition (Kashiwara [18]) The crystal graph of L(Λ) is the edge la-

belled directed graph with vertex set B_{0}(Λ) and edges b ↠^{i} b′ whenever F̃_{i}b = b′ for b, b′ ∈ B_{0}(Λ) and i ∈ I_{ℓ}.

Kashiwara proved that B_{0}(Λ) is a connected graph. Consequently, from Theorem 2.8 we obtain the following result.

2.10 Corollary The crystal graph of L(Λ) is the graph with vertices the set

𝒦 = ⋃_{n≥0} 𝒦_{n} of Kleshchev multipartitions and edges λ ↠^{i} μ given by adjoining good nodes.

Kashiwara and Lusztig have shown that B_{0}(Λ) lifts to the canonical basis (or global basis) of L(Λ) = L(Λ)_{A} ⊗_{A} ℂ(v), so we can specialize the canonical basis to give a basis of 𝔲_{R}^{0} indexed by B_{0}(Λ). This shows that, under the assumptions of (1.8), the Kleshchev multipartitions 𝒦_{n} index the simple modules of ℋ_{R,n} . Thus, by Theorem 1.3, we have proved Theorem C.

Jimbo et al [16] (see also [10]) have given a different description of the crystal graph of L(Λ). The point of this construction is the following conjecture.

2.11 Conjecture Suppose that R is a field, q ≠ 1 and that all of the parameters Q_{1}, …, Q_{r} are non–zero. Then D_{R}^{λ} ≠ 0 if and only if λ is a Kleshchev multipartition.

This conjecture is known to be true when r = 1 [7] and when q = −1 [22]. Graham and Lehrer [13] have given sufficient conditions for D_{R}^{λ} to be non– zero and they have conjectured that these conditions are also necessary. It seems likely that the Kleshchev multipartitions are precisely the multipartitions which satisfy the Graham–Lehrer conjecture. ^{1}

It remains to prove Theorem D. Recall the Weyl–Kac q–dimension formula [17, Proposition 10.10]. Since the positive root system for A_{ℓ−1}^{(1)} is explicitly known [17, Exercise 6.5], this implies Theorem D when ℓ < ∞. The case A_{∞} is also obvious.

3 Applications to the groups of Lie type

In this final section, we comment on the application of our results to the classification of the irreducible representations of finite classical groups in non–defining characteristic. In general, these modules are parameterised by triples (L, X, ϕ) where L is a Levi subgroup of G, X a cuspidal simple module of L and ϕ is a simple module of the endomorphism ring End_{G}(R_{L}^{G}(X)) of the Harish–Chandra induced module R_{L}^{G}(X); see [12, Theorem 2.4].

Assume that G is one of the following groups: GU_{n}(q), Sp_{2n}(q), CSp_{2n}(q) or SO_{2n+1}(q). Then Geck, Hiß and Malle [12, Proposition 4.4] have shown that End_{G}(R_{L}^{G}(X)) is isomorphic to a product of Hecke algebras of type B. Thus, given a cuspidal simple module, by determining the appropriate parameters for the associated Hecke algebra, our Theorem D gives the exact number of simple modules in the Harish–Chandra series of (L, X). Since the total number of simple G–modules is known [11] it is therefore sufficient to find enough cuspidal simple modules in order to complete the classification of the simple G–modules. This is not possible using the classification of the simple ℋ_{R,n} –modules proposed by Graham and Lehrer [13], because their classification does not give a way of counting the number of simple modules.

Thus, Theorem D solves problem (b) from the introduction of [12], modulo determination of certain parameter values. We can therefore focus on problem (a) of [12]; that is, the problem of finding enough cuspidal simple modules and determining the parameters for them.

Another application of our results is to the classification of the irreducible admissible R–representations of the general linear groups over p–adic fields. If R is a field of zero, this was completed by Bushnell and Kutzko [4]. Vigneras considered the case where the characteristic of R is different from that of the residue field. In [25], she reduced this problem to the classification of both the cuspidal R–representations of the p–adic general linear groups and also the irreducible representations of affine Hecke algebras of type A. Vigneras classified the cuspidal R–representations in [26]; she also conjectured [27] that the modular Deligne–Langlands parameters parametrise the unipotent admissible duals. Hence, Theorem B provides the final step in this classification.

1Conjecture 2.11 has recently been established by the first author [2]; moreover, the sets of Kleshchev multipartitions and those defined by Graham and Lehrer coincide.

Acknowledgements

The authors are grateful to M-F. Vigneras for explaining her work to us [28]. We also thank R. Dipper for several conversations which motivated some of the results in section 1.

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S.A. Tokyo University of Mercantile Marine, Etchujima 2-1-6,

Koto-ku, Tokyo 135-8533 Japan. ariki@ipc.tosho-u.ac.jp

A.M. University of Sydney, Sydney NSW 2006. Australia.

       a.mathas@maths.usyd.edu.au