QUANTUM WEYL RECIPROCITY FOR COHOMOLOGY

BRIAN J. PARSHALL AND LEONARD L. SCOTT

Abstract. We study the relationship between the cohomology of q-Schur algebras and Hecke algebras in type A. A key tool, which has independent interest, involves a specific Hecke algebra resolution, introduced in a special case by Deodhar [Deo]. This resolution is somewhat akin to the Koszul resolution for Lie algebra cohomology, in that it has an explicit description yet applies for all modules. An application of the properties of this resolution establishes a “quantum Weyl reciprocity” connecting, in a range of degrees, the cohomology of the q-Schur algebra with that of the Hecke algebra. This work also provides, independently of the resolution, cohomological inequalities which apply in all degrees and which sometimes can be related to Kazhdan-Lusztig polynomials. Finally, we extend to cohomology a decomposition number method of Erdmann [E], and illustrate its usefulness in computations. This method enables us to exhibit some properties of symmetric group cohomology which, if true, would imply the SL_{n}-version of Lusztig’s characteristic p character conjecture.

Contents

1. Preliminaries. 5 1.1. The ATR set-up. 5 1.2. Quasi-hereditary algebras. 7 2. Hecke algebras and q-Schur algebras. 9 2.1. Preliminaries on Hecke algebras. 9 2.2. Modular representation theory. 10 3. Resolutions. 15 3.1. The complex. 15 3.2. Passing to the q-Schur algebra. 19 4. Cohomology, I: isomorphisms. 23 5. Cohomology, II: inequalities. 26 6. Cohomology, III: smaller q-Schur algebras and specific calculations. 29 7. Appendix: the Lusztig conjecture and the symmetric group. 33 References 34

Classical Weyl reciprocity relates the representation theories of the general linear group GL_{n}(k) and the symmetric group 𝔖_{r} by means of the (GL_{n}(k), 𝔖_{r})-bimodule T := V^{⊗r}, where V = k^{n}. For example, when k = ℂ, the poset Λ^{+}(n, r) of partitions of r with at most n nonzero parts indexes the irreducible polynomial GL_{n}(ℂ)-modules

1991 Mathematics Subject Classification. Primary 17B55, 20G; Secondary 17B50. Research supported in part by the National Science Foundation.

L(λ), which are homogeneous of degree r, and a subset of the complex irreducible 𝔖_{r}- modules S_{λ}. As a bimodule, T decomposes into a direct sum of bimodules L(λ) ⊗ S_{λ}.

In quantum Weyl reciprocity, GL_{n}(k) is replaced by the quantum enveloping algebra U_{q} of type A_{n−1} (over an arbitrary algebraically closed field k), while 𝔖_{r} is replaced by the Hecke algebra H = H(𝔖_{r}) of type A_{r−1}. Further, q-tensor space T_{q} serves as a natural bimodule for the pair (U_{q}, H), and the image U_{q} → End_{H}(T_{q}) defines the q-Schur algebra S_{q}(n, r). If k = ℂ and q is not a root of unity, results of Jimbo largely recover results analogous to the classical q = 1 case. In general, T_{q} is not completely reducible for the action of S_{q}(n, r) or for that of H. However, in analogy with the classical results, there is a filtration of T_{q} with sections of the form Δ(λ) ⊗ S_{λ}, λ ∈ Λ^{+}(n, r) [DPS2]. Here, Δ(λ) denotes the standard module for U_{q} of highest weight λ, and S_{λ} is the Specht module for H corresponding to λ. The category of left S_{q}(n, r)- modules and that of right H-modules are related by the contravariant functors

\begin{cases}Hom_{S_{q}(n,r)}(−, T) : S_{q}(n, r)–mod → mod–H, \\ Hom_{H}(−, T) : mod–H → S_{q}(n, r)–mod \end{cases} (0.0.1)

which have been studied extensively in [CPS4], [DPS1], [DPS2]. In particular, when r ≤ n, T ≅ S_{q}(n, r)e and H^{op} ≅ eS_{q}(n, r)e, for some idempotent e ∈ S_{q}(n, r). In this case, Hom_{S_{q}(n,r)}(−, T) is closely related (by duality) to the famous Schur functor

F : S_{q}(n, r)–mod → H–mod, M ↦ eM. (0.0.2)

This paper extends quantum Weyl reciprocity to include cohomology by investigating a Hecke algebra complex, somewhat in spirit at least to the classical Koszul complex for Lie algebra cohomology. The complex provides an explicit resolution

0 → Φ → 𝔑_{0} → ⋯ → 𝔑_{r−1} → 0 (0.0.3)

of the functor Φ which carries an H-module to its “twist” by the sign representation SGN of H. When (0.0.3) is evaluated at the trivial representation for H, Deodhar [Deo] proved the resulting complex provides a resolution

0 → SGN → N_{0} → ⋯ → N_{r−1} → 0 (0.0.4)

of the sign representation for H. In turn, (0.0.4) determines, via Weyl reciprocity and tilting (at the integral level), a resolution of the quantum determinant det_{q} for the q-Schur algebra, providing an essential link connecting the groups Ext_{U_{q}}^{•}(det_{q}, −) = Ext_{S_{q}(n,n)}^{•}(det_{q}, −) with corresponding Ext^{•}–groups for the finite general linear groups GL_{n}(𝔽) in which 𝔽 has characteristic distinct from the characteristic of the field k—cf. [CPS5]. (The “q” in S_{q}(n, r) is the image in k of the integer |𝔽|.)

A main result of this paper shows that (0.0.3) leads to cohomological quantum Weyl reciprocity results. For example, when r ≤ n, there is an isomorphism

Ext_{S_{q}(n,r)}^{•}(M, Δ(λ)) ≅ Ext_{H}^{•}(S_{λ}, M^{⋄}), (0.0.5) valid in a range of cohomological degrees, for any finite dimensional S_{q}(n, r)-module M and any partition λ of r with at most n nonzero parts. Here, M^{⋄} = (FM)^{∗}, the linear dual of the image of M under the Schur functor F. A similar result in the q = 1 case had been established by Kleshchev and Nakano [KN]. For q ≠ 1 and cohomological degree 1, there are some related results in [DEN] and [HN].

Our paper is organized as follows. Section 1 collects together some general facts which will be necessary later. Hecke algebras and q-Schur algebras make their appearance in Section 2, which establishes several cohomological properties of Specht and Young modules. Throughout the paper, an approach at the integral level over a discrete valuation ring plays an important role. In this sense, §§1,2 depend upon the elementary integral representation theory developed in [CPS4] and [DPS1].

Section 3 provides an investigation into the resolution (0.0.3), whose existence is established in Thm. 3.4. The technical, but important, Thm. 3.3 shows that (0.0.3) behaves well with respect to the Mackey decomposition theory for Hecke algebras. We then apply these results to understand how (0.0.3), when evaluated at Young, twisted Young, and Specht modules, dualizes to the category of modules for the q-Schur algebra.

While these results have independent interest, they can easily be applied, as we illustrate in Section 4, to obtain the cohomological results along the lines of (0.0.5) mentioned above. An interesting fact, which plays an important role, establishes that the groups Ext_{S_{q}(r,r)}^{i}(∇(λ), S_{q}(r, r)) vanish in a range of degrees i > 0. Here ∇(λ) is the costandard S_{q}(r, r)-module, dual to Δ(λ); cf. (4.4). Another result shows that Ext_{S_{q}(r,r)}^{i}(∇(λ), L(μ)) = 0 in a range of degrees provided that μ is not restricted. This fact gives a cohomological and quantum generalization of a well-known result of James [J] which states, in the case i = 0 and q = 1, that the head of any ∇(λ) consists of irreducible modules L(μ) with restricted weights μ.

In general, S_{q}(r, r) has finite global dimension, while H has infinite global dimension. Hence, we must expect that any comparison between cohomology groups will only hold in a range of cohomological degrees i. Generally, that range is i ≤ e−2, for e > 2, with e as in (2.0.17). However, §5 establishes, that, for any S_{q}(r, r)-module M and any i ≥ 0, there is, subject to a hypothesis on the vanishing of Ext_{S_{q}(r,r)}^{j}(M, L(ω)) for certain partitions ω and j < i, an inequality dim Ext_{S_{q}(r,r)}^{i}(M, L(ν)) ≤ dim Ext_{H}^{i}(FM, D_{ν′}^{Φ}) for any restricted partition ν. (D_{ν′} is the irreducible head of S_{ν′}.) This inequality yields cohomology estimates which cannot, in general, be handled by the methods of §4.

These results use the q-Schur algebra S_{q}(r, r). When n ≥ r, S_{q}(n, r) is Morita equivalent to S_{q}(r, r), so that similar results hold here. Section 6 shows how the preceding results apply when n < r by addressing the question of transferring cohomological information from the S_{q}(n, r)–case to the S_{q}(r, r)–case, making the preceding results more useful. For example, when q = 1 and H = k𝔖_{r}, if p < r the Lusztig mod p character conjecture [L] does not apply to S(r, r) = S_{q}(r, r), while it does apply to S(n, r) whenever n ≤ p. In this case, cohomology calculations for S(r, r) can be sometimes reduced to similar calculations for S(n, r) and then solved in terms of the coefficients of certain Kazhdan-Lusztig polynomials. The main tool is Thm. 6.1, which, in particular, relates Ext_{S_{q}(n,r)}^{i}(Δ(λ), L(μ)) with the similar expression for the algebra S_{q}(r, r). The methods of Section 5 yield interesting inequalities on symmetric group cohomology. We include an illustration of this technique for the group GL_{5}(k), when k has characteristic 7—a highly non-trivial situation in which the Lusztig conjecture has been verified [S]. This example is based on an interesting Ext-transfer result for Schur algebra cohomology which can viewed as a natural cohomological extension of Erdmann’s results on decomposition numbers [E]. In effect, non-p-restricted weights are converted to weights that are p-restricted for a larger r. Finally, in a very brief Appendix, §7, we show how these results lead to some properties of symmetric group cohomology which, if true, would imply Lusztig’s conjecture in the special case of SL_{n}(k).

In writing this paper, especially §4, the authors derived inspiration from earlier work of Kleshchev and Nakano [KN], which studied the relationship between the cohomology of the symmetric group and that of the classical Schur algebra. However, our approach even in this q = 1 case is quite different, because it is based on properties of the functor resolution (0.0.3). In particular, it does not depend on cohomology calculations for p-Sylow subgroups—not presently available in the general q–context—or on Kleshchev’s theory of completely splittable weights. In addition, a few of our results—mostly in the last three sections—are even new in the q = 1 case.

By Dipper-James theory, q-Schur algebras play a central role in the non-describing characteristic representation theory of the finite groups GL_{n}(𝔽_{q}):

GL_{n}(𝔽_{q})

[figure: Dipper-James theory]
S_{q}(n, n)

Weyl reciprocity

H

Our paper provides a cohomological interpretation of the base of this triangle. In a sequel, we will apply these results to further understand cohomologically its left side.

Some notation. Throughout this paper, Z := ℤ[t, t^{−1}] denotes the ring of integer Laurent polynomials in a variable t. If A is a ring, let A–mod (resp., mod–A) denote the category of left (resp., right) finitely generated A-modules. In practice, A will be an algebra over a commutative ring Z; most often, Z will be a field, a discrete valuation ring (DVR), or the ring Z above. If M is a Z-module, M^{∗} denotes Hom_{A}(M, Z).

A local triple (O, K, k) consists of a commutative local domain O with fraction field K and residue field k = O/𝔪. If M̃ is a O-module, write M̃_{K} = K ⊗_{O} M̃ and M̃_{k} = k ⊗_{O} M̃ = M̃/𝔪M̃. The following notational convention will be used: An O-module will often be denoted M̃, while M denotes M̃_{k}.

If C is a class in A-mod and M ∈ A–mod, write M ∈ A–mod(C), or say “M has a C-filtration”, if M has a filtration 0 = M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{t} = M with each section M_{i}/M_{i−1} isomorphic to an object in C. If A is a quasi-hereditary algebra, let C = Δ (resp., ∇) be the class of standard (resp., costandard) modules Δ(λ) (resp., ∇(λ)).

Let Λ^{+}(n, r) be the set of partitions λ of r (notation: λ ⊢ r) having at most n nonzero parts: λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n} ≥ 0), |λ| := λ_{1}+⋯+λ_{n} = r. If λ ∈ Λ^{+}(n, r), set λ_{m} = 0 for m ≥ n, so that λ may be viewed as an infinite non-increasing sequence of nonnegative integers which is eventually 0. Put Λ^{+}(r) = ⋃_{n≥0} Λ^{+}(n, r) = Λ^{+}(r, r). The set Λ^{+}(r) is a poset, putting, for λ, μ ∈ Λ^{+}(r), λ ⊴ μ provided λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i} for all i. Each Λ^{+}(n, r) is a coideal (i. e., λ ⊵ μ ∈ Λ^{+}(n, r) ⟹ λ ∈ Λ^{+}(n, r)). Given λ ∈ Λ^{+}(r), let λ′ ∈ Λ^{+}(r) be the dual partition, i. e., λ′_{i} = #{λ_{j} ≥ i}, i = 1, 2, . . ..

Call λ ⊢ r e-regular for a positive integer e provided that no nonzero part λ_{i} is repeated e or more times. Let Λ_{e-reg}^{+}(r) be the set of e-regular partitions, and put Λ_{e-reg}^{+}(n, r) = Λ_{e-reg}^{+}(r) ∩ Λ^{+}(n, r). If λ′ ∈ Λ_{e-reg}^{+}(r), then λ is called e-restricted. Let Λ_{e-res}^{+}(r), Λ_{e-res}^{+}(n, r) be the corresponding sets of e-restricted partitions.

Let S = {(1, 2), ⋯, (r − 1, r)} be the set of fundamental reflections for 𝔖_{r}; thus (𝔖_{r}, S) is a Coxeter system. For J ⊆ S, let 𝔖_{J} be the Young subgroup of 𝔖_{r} generated J. It is convenient to associate Young subgroups to partitions: given λ ⊢ r, put 𝔖_{λ} = 𝔖_{J(λ)}, where J(λ) = {(1, 2), ⋯, (λ_{1}−1, λ_{1}), (λ_{1}+1, λ_{1}+2), ⋯, (λ_{2}−1, λ_{2}), ⋯ }.

Given a functor F : A → B and X ∈ A, we sometimes use “exponential” notation X^{F} to denote the image F (X) ∈ B.

1. Preliminaries.

1.1. The ATR set-up. Let Ã, R̃ be algebras over a commutative ring Z and let T̃ be an (Ã, R̃)-bimodule T̃. We make the blanket assumption: Ã, T̃, R̃ are free Z-modules of finite rank. The contravariant ‘diamond’ functors

(−)^{⋄} = HomÃ(−, T̃) : Ã–mod → mod–R̃;
(−)^{⋄} = HomR̃(−, T̃) : mod–R̃ → Ã–mod.

(1.0.6)

play an important role. Evaluation defines natural transformations

(1.0.7) Ev : Id_{Ã–mod} → (−)^{⋄⋄}; Ev : Id_{mod–R̃} → (−)^{⋄⋄}.

We always assume that the right R̃-module T̃ has a direct sum decomposition

(ATR1)  T̃R̃ = ⨁_{λ∈Λ} Ỹ_{λ}^{⊕m_{λ}},

where the Ỹ_{λ} are distinct indecomposable R̃-modules. We will assume that in any other decomposition of T̃R̃ into indecomposable summands, the indecomposable summands are isomorphic to the Ỹ_{λ} with the same multiplicities. In practice, the indexing set Λ will be a poset. In addition, assume that

(ATR2)  Ã ≅ EndR̃(T̃R̃),

as algebras. Condition (ATR1) always holds if Z if a field k. But it also holds if Z is a DVR with residue field K so that R̃_{K} is a split semisimple K-algebra, since, in this case, the Krull-Schmidt theorem holds for mod–R̃ (Heller’s theorem). Since A_{K} ≅ EndR̃_{K}(T̃_{K}), Ã_{K} is also a split semisimple K-algebra, so that the Krull-Schmidt theorem holds for Ã–mod as well. Finally, let A = Ã_{k}, T = T̃_{k} and R = R̃_{k}. Assume that the isomorphism Ã →^{∼} EndR̃(T̃) remains an isomorphism A →^{∼} End_{R}(T) after base change to k (which holds if and only if Ext1 ; R̃(T̃, T̃) = 0 [CPS4, (1.5.2e,f)]). Then the Y_{λ} := Ỹ_{λk}, λ ∈ Λ, are the distinct indecomposable summands of T (each occurring with the same multiplicity as the Ỹ_{λ} do in T̃); cf. [CPS4, (1.5.6b)].

The modules {P̃(λ) = Ỹ_{λ}^{⋄}}_{λ∈Λ}, are the distinct projective indecomposable modules for Ã–mod. Also, the evaluation maps (1.0.7) Ev_{P̃(λ)} : P̃(λ) →^{∼} P̃(λ)^{⋄⋄} and EvỸ_{λ} : Ỹ_{λ} →^{∼} Ỹ_{λ}^{⋄⋄} are isomorphisms. Similar remarks hold for the P (λ) = P̃(λ)_{k} and Y_{λ}.

Remark 1.1. Consider algebras over a fixed commutative ring Z. Assume that R̃R̃ is a direct summand of T̃R̃. Then T̃ ≅ Ãe in Ã–mod for some idempotent e ∈ Ã; see [CPS4, (1.1.11)], where the argument, given in the field case, works equally well with the field replaced by Z. Also assume that R̃^{∗} ≅ R̃ as (R̃, R̃)-bimodules (i. e., R̃ is Frobenius). Then (Ae)^{∗} ≅ eA as (R̃, Ã)-bimodules—cf. the argument in [CPS4, (1.1.11.3)]. Then, if M̃ ∈ Ã–mod is Z-projective,

M̃^{⋄} ≅ HomÃ(M̃, Ãe) ≅ HomÃ((Ãe)^{∗}, M̃^{∗})
≅ HomÃ(eÃ, M̃^{∗}) ≅ M̃^{∗}e ≅ (eM̃)^{∗},

as right R̃-modules. The functor F : Ã–mod → R̃–mod, M̃ ↦ eM̃, is called the Schur functor. Here we identify R̃ ≅ R̃^{op} with eÃe. The functorial isomorphism

(1.1.1) FM̃ ≅ M̃^{⋄∗}, M̃ ∈ Ã–mod, Z-projective

relates the Schur and ‘diamond’ functors.

Assume Z = O is a DVR. Then F–mod has right adjoint G = Hom_{eÃe}(eÃ, −) : R̃–mod → Ã–mod such that the adjunction transformation FG →^{∼} Id is an isomorphism. Thus, G is a full embedding. Also, EvÑ : Ñ →^{∼} Ñ^{⋄⋄} is an isomorphism for all O-free R̃-modules Ñ: by Prop. 1.2 below, G(Ñ) ≅ Ñ^{∗⋄}, if Ñ is a O-free left R̃-module. Then EvÑ is the linear dual of the adjunction isomorphism FG(Ñ) →^{∼} Ñ.

Proposition 1.2. Assume that O is a DVR. Let M̃ ∈ Ã–mod and Ñ ∈ mod–R̃. Assume Ñ is O-free. Then

HomÃ(M̃, Ñ^{⋄}) ≅ HomR̃(Ñ, M̃^{⋄}),

functorially in M̃ and Ñ.

Proof. First, the natural map T^{∗} ⊗Ã M̃ → HomÃ(M̃, T̃)^{∗} is an isomorphism in R̃–mod. This is obvious if M̃ is Ã-free, and follows generally since the functors T̃^{∗} ⊗Ã − and HomÃ(−, T̃)^{∗} are both right exact. Then

HomÃ(M̃, Ñ^{⋄}) ≅ HomÃ(M̃, HomR̃(Ñ, T̃)) ≅ HomÃ(M̃, HomR̃(T̃^{∗}, Ñ^{∗}))
≅ HomR̃(T̃^{∗} ⊗Ã M̃, Ñ^{∗}) ≅ HomR̃(M̃^{⋄∗}, Ñ^{∗})
≅ HomR̃(Ñ, M̃^{⋄}),

since M̃^{⋄} is automatically O-free, □

For simplicity, the following application is given only over a field k.

Corollary 1.3. In Prop. 1.2 assume that O = k is a field, and that T is an injective A-module.

(a) For M ∈ A–mod, N ∈ mod–R, there is a (Grothendieck) spectral sequence

E_{2}^{p,q} = Ext_{A}^{p}(M, Ext_{R}^{q}(N, T)) ⟹ Ext_{R}^{p+q}(N, M^{⋄}).

(b) Assume that Ext_{R}^{q}(N, T) = 0, for 0 < q ≤ s. There exists a 5-term exact sequence

0 → Ext_{A}^{s+1}(M, N^{⋄}) → Ext_{R}^{s+1}(N, M^{⋄}) → Hom_{A}(M, Ext_{R}^{s+1}(N, T))
→ Ext_{A}^{s+2}(M, N^{⋄}) → Ext_{R}^{s+2}(N, M^{⋄}).

Proof. Fix M. The functor factorization Hom_{R}(−, M^{⋄}) = Hom_{A}(M, −) ∘ Hom_{R}(−, T) holds by Prop. 1.2. If Q ∈ mod–R is projective (i. e., injective in the opposite category mod–R^{op}), Q^{⋄} = Hom_{R}(Q, T) is a direct summand of some T^{⊕m} and so, by hypothesis, injective for A. Thus, the spectral sequence in (a) exists. Finally, (a) ⟹ (b). □

The reader may compare (1.3a) to the complementary covariant spectral sequence [DEN, (2.4)], which compares Ext^{•}-groups for an algebra A with those for R = eAe, e ∈ A an idempotent, using the Schur functor F above (dropping the tilde notation in (1.1)). This spectral sequence predates [KN], and was heavily used in the latter. However, this eAe-setup does not give the appropriate relationship between Hecke and q-Schur algebras, discussed in §2, unless n ≥ r. But Lemma 1.3(a) applies for all nonnegative integers n.

1.2. Quasi-hereditary algebras. Let A be a quasi-hereditary algebra (QHA) over an algebraically closed field k; so A–mod is a highest weight category (HWC) with respect to a poset structure ≤ on a “weight” set Λ indexing the distinct irreducible objects L(λ) = L_{A}(λ) ∈ A–mod. For λ ∈ Λ, let Δ(λ) = Δ_{A}(λ) (resp., ∇(λ) = ∇_{A}(λ)) be the standard (resp., costandard) object in A–mod with head (resp., socle) L(λ). An A-module X is called a tilting module if it has both a Δ– and a ∇–filtration. For λ ∈ Λ, there exists a unique indecomposable tilting module X(λ) such that λ is maximal among those μ ∈ Λ for which Δ(μ) (resp., ∇(μ)) is a Δ–section (resp., ∇–section) in X(λ); cf. [R].

We require some basic results concerning integral quasi-hereditary algebras [CPS2]. Let Ã be a QHA over a DVR O with quotient field K and algebraically closed residue field k. Assume that Ã is a free O-module of finite rank and that Ã_{K} is a split semisimple algebra. Also, A = Ã_{k} is a QHA over k with poset denoted Λ. The category Ã–mod is an integral HWC relative with poset Λ. For each λ ∈ Λ, there exist O-free Δ̃(λ) (resp., ˜∇(λ)) in Ã–mod such that Δ(λ) := Δ̃(λ)_{k} (resp., ∇(λ) := ˜∇(λ)_{k}) are the standard (resp., costandard) objects in A–mod with respect to the poset Λ.^{1} In addition, the Δ̃(λ)_{K} ≅ ˜∇(λ)_{K}, λ ∈ Λ, are the distinct irreducible Ã_{K}-modules. The following is well-known. (For (b), use [CPS4, (4.5.1)].)

Lemma 1.4. Let Ã be a QHA over O as above. Then:

(a) For λ, μ ∈ Λ, dim Ext_{A}^{i}(Δ(λ), ∇(μ)) = δ_{i,0}δ_{λ,μ}. Thus,

Exti ; Ã(Δ̃(λ), ˜∇(μ)) ≅ \begin{cases}O, i = 0, λ = μ; \\ 0, i > 0 or λ ≠ μ. \end{cases} (1.4.1)

(b) Let 0 → M̃_{n} → ⋯ → M̃_{0} → 0 be an exact sequence in Ã–mod with M̃_{i} ∈ Ã–mod(Δ̃) for 0 ≤ i < n. Then M̃_{n} ∈ Ã–mod(Δ̃). The analogous statement holds in A–mod. (A dual statement holds for Ã–mod( ˜∇), etc.)

As with A–mod, any X̃ ∈ Ã–mod(Δ̃) ⋂ Ã–mod( ˜∇) is called a tilting module. By [CPS4, 4.5.2], given λ ∈ Λ, there exists O-free tilting module X̃(λ) ∈ Ã–mod such that X̃(λ)_{k} ≅ X(λ), the indecomposable tilting module in A–mod of highest weight λ.

Lemma 1.5. (a) X̃(λ) is the unique O-free module X̃ in Ã–mod satisfying X̃_{k} ≅ X(λ).

(b) Any Ỹ ∈ Ã–mod(Δ̃) has a finite resolution 0 → Ỹ → X̃_{0} → ⋯ → X̃_{n} → 0 such that each X̃_{i} is a tilting module and each kernel Ker(X̃_{i} → X̃_{i+1}) has a Δ̃-filtration.

Proof. (a) is an immediate application of [CPS4, (1.5.2b,c,e)] and Nakayama’s lemma. To see (b), consider (Ã, T̃, R̃), where T̃ = ⨁_{λ} X̃(λ) and R̃ = EndÃ(T̃)^{op}. Then Ã is the Ringel dual of the QHA R̃ and T̃ is a tilting module for R̃. For λ ∈ Λ, Δ̃R̃(λ)^{⋄} ≅ Δ̃Ã(λ) and Δ̃Ã(λ)^{⋄} ≅ Δ̃R̃(λ) [CPS4, §§3.4, 4.3]. Now choose a finite projective resolution

P̃_{•} → Ỹ^{⋄} → 0. (1.5.1)

By Lemma 1.4(b), the kernels (or cokernels) in this resolution have Δ̃-filtrations. Then (−)^{⋄} = HomR̃(−, T̃) carries each P̃_{i} into a tilting module for Ã. By [CPS4, (4.3.1)], EvỸ : Ỹ →^{∼} Ỹ^{⋄⋄} is an isomorphism. Thus, by Lemma 1.4(a), the desired resolution of Ỹ is obtained by applying (−)^{⋄} = HomR̃(−, T̃) to (1.5.1). □

1By [CPS4, §4.5], Δ̃(λ) (resp., ˜∇(λ)) is the unique O-free object in Ã–mod lifting Δ(λ) (resp., ∇(λ)).

QUANTUM WEYL RECIPROCITY FOR COHOMOLOGY 9

2. Hecke algebras and q-Schur algebras.

2.1. Preliminaries on Hecke algebras. Let (W, S) be a finite Coxeter system. The Hecke algebra H̃ = H̃(W, S) is the free Z-algebra with basis τ_{w}, w ∈ W, and relations

(2.0.2) τ_{s}τ_{w} = \begin{cases}τ_{sw}, if ℓ(sw) = ℓ(w) + 1; \\ tτ_{sw} + (t − 1)τ_{w}, otherwise. \end{cases}

Here ℓ(w) denotes the length of w ∈ W with respect to S. There is a Z-involution

Φ : H̃ → H̃, τ_{w} ↦ (−t)^{ℓ(w)}τ_{w^{−1}}^{−1}, ∀w ∈ W. (2.0.3)

If M̃ is an H̃-module, then M̃^{Φ} denotes the new H̃-module obtained from M̃ by twisting the action of H̃ by Φ. This defines an exact functor

(2.0.4) Φ : H̃–mod → H̃–mod, M̃ ↦ M̃^{Φ}.

A similar functor, also denoted Φ, is defined on mod–H̃. Given M̃ ∈ H̃–mod (resp., mod–H̃), the right (resp., left) module M̃^{∗} can be converted into a left (resp. right) module M̃^{DH̃} by means of the anti-automorphism ι : H̃ → H̃, ι(τ_{w}) = τ_{w^{−1}}. If M̃ is Z-free, then M̃^{D2 ; H̃} ≅ M̃. In addition, ι defines a covariant isomorphism

ι : mod–H̃ ⟶^{∼} H̃–mod (2.0.5)

of categories: if M̃ is a right H̃-module, M̃^{ι} is the corresponding left H̃-module obtained by making H̃ act through ι.

The algebra H̃ has linear characters ĨND and { \atop SGN}, defined by: ĨND : H̃ → Z,

τ_{w} ↦ t^{ℓ(w)} and { \atop SGN} : H̃ → Z, τ_{w} ↦ (−1)^{ℓ(w)}. Then

(2.0.6) \begin{cases}ĨND^{Φ} ≅ { \atop SGN}; \\ { \atop SGN}^{Φ} ≅ ĨND; \end{cases} and \begin{cases}ĨND^{DH̃} ≅ ĨND; \\ { \atop SGN} & DH̃ & ≅ & { \atop SGN} \end{cases} .

For J ⊆ S, let W_{J} be the parabolic subgroup of W generated by the J. Let D_{J} be the set of distinguished right coset representatives of W_{J} in W, so that W_{J} ×D_{J} ⟶^{mult} W is a bijective map such that ℓ(wd) = ℓ(w) + ℓ(d) for w ∈ W_{J}, d ∈ D_{J}. Let H̃_{J} be the Hecke algebra associated to the Coxeter system (W_{J}, J). Observe that H̃ is a free left (resp., right) H̃_{J}-module with basis τ_{d}, d ∈ D_{J} (resp., d ∈ D_{J}^{−1}).

We recall some properties of induction functors, working with right modules to be explicit. The restriction functor ResH̃ ; H̃_{J} : mod–H̃ → mod–H̃_{J} admits a left adjoint

(2.0.7) Ind^{⊗}H̃ ; H̃_{J} = − ⊗H̃_{J} H̃ : mod–H̃_{J} → mod–H̃ and a right adjoint

(2.0.8) IndH̃ ; H̃_{J} = HomH̃_{J} (H̃, −) : mod–H̃_{J} → mod–H̃ Because H̃ is a free left H̃_{J}-module, both adjoints are exact functors.

If J ⊆ K ⊆ S and M̃ ∈ mod–H̃_{K}, the identity map M̃ → M̃ (viewed as a morphism of mod–H̃_{J}-modules), uniquely extends to a morphism Ind^{⊗}H̃_{K} ; H̃_{J} ∘ ResH̃_{K} ; H̃_{J}(M̃) → M̃ and, then, to a morphism

(2.0.9) Ind^{⊗}H̃ ; H̃_{J} ∘ ResH̃_{K} ; H̃_{J}(M̃) → Ind^{⊗}H̃ ; H̃_{K}(M̃)

of H̃-modules, explicitly described as m ⊗H̃_{J} h ↦ m ⊗H̃_{K} h, ∀h ∈ H̃, m ∈ M̃. If we start with a H̃-module M̃, we obtain a similar map (keeping the same notation)

(2.0.10) ϕ_{J,K}(M̃) : Ind^{⊗}H̃ ; H̃_{J} ∘ ResH̃ ; H̃_{J} (M̃) → Ind^{⊗}H̃ ; H̃_{K} ∘ ResH̃ ; H̃_{K}(M̃).

Clearly, ϕ_{J,K} is a natural transformation of functors.

If M̃ ∈ mod–H̃ is a Z-free module, then

(Ind^{⊗}H̃ ; H̃_{J} M̃)^{DH̃} ≅ IndH̃ ; H̃_{J} (M̃^{DH̃_{J}}). (2.0.11)

If M̃ = ĨND, then (Ind^{⊗}H̃ ; H̃_{J} ĨND)^{DH̃} ≅ Ind^{⊗}H̃ ; H̃_{J} ĨND, so that

(2.0.12) IndH̃ ; H̃_{K} ∘Ind^{⊗}H̃_{K} ; H̃_{J} ĨND ≅ Ind^{⊗}H̃ ; H̃_{J} ĨND.

Given J, K ⊆ S, put D_{K,J} = D_{K} ∩ D_{J}^{−1}. For d ∈ D_{K,J}, K^{d} ∩ J = d^{−1}Kd ∩ J ⊆ S (Kilmoyer’s theorem). Let M̃ ∈ mod–H̃_{K}. For d ∈ D_{K,J}, let M̃ ⊗ τ_{d} ∈ mod–H̃_{K^{d}∩J} be defined by the action (m ⊗ τ_{d})τ_{u} := mτ_{v} ⊗ τ_{d}, m ∈ M̃, u = d^{−1}vd ∈ K^{d} ∩ J. Then

(2.0.13) ResH̃ ; H̃_{J} ∘Ind^{⊗}H̃ ; H̃_{K} M̃ ≅ ⨁_{d∈D_{K,J}} Ind^{⊗}H̃_{J} ; H̃_{K^{d}∩J} M̃ ⊗ τ_{d}.

This is the Mackey decomposition—see [GP, (9.1.8)].

2.2. Modular representation theory. Let H̃ = H̃(W, S) be the Hecke algebra over Z with W = 𝔖_{r} and S == {(1, 2), (2, 3), ⋯, (r − 1, r)}. Let k be an algebraically closed field of characteristic p ≥ 0. Assume that k is a Z-algebra by means of a fixed ring homomorphism Z → k, t → t̄. Put q = t̄ and write H = H̃_{k}. Then H is not semisimple if and only if t̄ is a root of the Poincaré polynomial

℘_{W} (ξ) := ∏_{i=1}^{r} \frac{ξ^{i} − 1}{ξ − 1},

cf., e. g., [DPS1, (4.2.2)]. Fix a Z-algebra O which is a DVR with quotient field K of characteristic 0 and residue field k. Assume the image of t in O is not a root of unity.

Integral Stratification Hypothesis and q-Schur algebras.

For λ ∈ Λ^{+}(r), put T̃_{λ} = Ind^{⊗}H̃_{O} ; H̃_{λO} ĨND ∈ mod–H̃_{O}. For a positive integer n, put

T̃(n, r) = ⨁_{λ∈Λ^{+}(n,r)} T̃_{λ}^{⊕n_{λ}}, (2.0.14)

for some choice of positive integers n_{λ}. (The specific values of the n_{λ} will not be important, since different choices just lead to Morita equivalent algebras.) Let T (n, r) := T̃(n, r)_{k}. The q-Schur algebra

S̃_{q}(n, r) := EndH̃_{O}(T̃(n, r)) (2.0.15)

satisfies S_{q}(n, r) := S̃_{q}(n, r)_{k} ≅ End_{H}(T (n, r)).^{2} The free O-algebra S̃_{q}(n, r) is an integral QHA over O with poset Λ^{+}(n, r), as discussed in Subsection (1.2). Further, S̃(n, r)_{K} ≅ EndH̃_{K}(T̃(n, r)_{K}) is a split semisimple algebra over K, since H̃_{K} is split semisimple. The triple (S̃_{q}(n, r), T̃(n, r), H̃_{O}) satisfies the conditions (ATR1,2) stated earlier. As an integral QHA, S̃_{q}(n, r) has standard objects Δ̃(λ) = S̃_{λ}^{⋄}, λ ∈ Λ^{+}(n, r), where {S̃_{λ}}_{λ∈Λ^{+}(r)} are the integral Specht modules which will be defined presently.

The Ext^{1}-vanishing criterion [DPS1, (2.3.9)] implies that (S̃_{q}(n, r), T̃(n, r), H̃_{O}) satisfies the Integral Stratification Hypothesis [CPS4, (4.1)], and so (S_{q}(n, r), T (n, r), H) satisfies the Stratfication Hypothesis [CPS4, Ch. 2]. For λ ∈ Λ^{+}(r),

\begin{cases}S̃_{λ}^{⋄} ≅ Δ̃(λ); \\ Δ̃(λ)^{⋄} ≅ S̃_{λ}; \\ ˜∇(λ)^{⋄} ≅ S̃_{λ′}^{Φ}, \end{cases} and \begin{cases}Δ(λ)^{⋄} ≅ S_{λ}; \\ ∇(λ)^{⋄} ≅ S_{λ′}^{Φ}. \end{cases} (2.0.16)

To see the assertion concerning ˜∇(λ)^{⋄}, observe that ˜∇(λ)^{⋄} ≅ Δ̃(λ)^{DÃ⋄} ≅ Δ̃(λ)^{⋄DH̃_{O}} ≅

S̃_{λ}^{DH̃_{O}} ≅ S̃_{λ′}^{Φ}, using (2.1.1) and [CPS3, (7.3)]. A similar argument establishes the claim for ∇(λ)^{⋄}. In general, it fails that S_{λ}^{⋄} ≅ Δ(λ), though see Lemma 2.4 below.

We will often concentrate on the triple (S̃_{q}(r, r), T̃(r, r), H̃), i. e., the case n = r. Then we usually write T̃ for T̃(r, r). In all cases, the arguments work for n ≥ r, and often they work for general n. Also, see §6. When n = r, T̃ ≅ S̃_{q}(r, r)e for an idempotent e ∈ S̃_{q}(r, r).

Young and twisted Young modules.

By [DPS2, §7], HomH̃_{O}(T̃_{λ′}^{Φ}, T̃_{λ}) ≅ O. Since the Krull-Schmidt theorem holds for finitely generated H̃_{O}-modules, there exist unique indecomposable summands Ỹ_{λ}^{♮} of T̃_{λ′}^{Φ} and Ỹ_{λ} of T̃_{λ} such that HomH̃_{O}(Ỹ_{λ}^{♮}, Ỹ_{λ}) ≅ O. The Ỹ_{λ} (resp., Ỹ_{λ}^{♮}), as well as the corresponding H-modules Y_{λ} = Ỹ_{λk} (resp., Y_{λ}^{♮} := Y_{λ′k}^{♮}) are called Young modules (resp., twisted Young modules). The Young modules Ỹ_{λ}, λ ∈ Λ^{+}(n, r), are precisely the indecomposable summands of T̃(n, r) as a right H̃_{O}-module. Thus,

2When q = 1, we denote S_{q}(n, r) simply by S(n, r)—these are the classical Schur algebras.

T̃(n, r) = ⨁_{λ∈Λ^{+}(n,r)} Ỹ_{λ}^{⊕r_{λ}}.

for positive integers r_{λ}. We also record that Ỹ_{λ}^{♮} ≅ Ỹ_{λ′}^{Φ}.

Specht and irreducible modules.

Let ζ_{λ} : Ỹ_{λ}^{♮} → Ỹ_{λ} be a generator for the free rank one O-module HomH̃_{O}(Ỹ_{λ}^{♮}, Ỹ_{λ}). Let S̃_{λ} denote the image of ζ_{λ}. The S̃_{λ}, λ ∈ Λ^{+}(r), are the Specht modules for H̃_{O}. The {S̃_{λK}}_{λ∈Λ^{+}(r)} form a complete set of distinct irreducible H̃_{K}-modules. The Specht modules for H are the modules S_{λ} := S̃_{λk} obtained by base change to k.

Assume that H is not semisimple. Thus, t̄ ∈ k is a primitive ℓth root of unity for some ℓ ≤ r. Let p ≥ 0 be the characteristic of k, and define

(2.0.17) e = \begin{cases}p if ℓ = 1; \\ ℓ otherwise. \end{cases}

The set Λ_{e-reg}^{+}(r) indexes the distinct irreducible H-modules as follows: given λ ∈ Λ_{e-reg}^{+}(r), let D_{λ} := S_{λ}/rad S_{λ} be the corresponding irreducible H-module. If λ ∉ Λ_{e-reg}^{+}(r), set D_{λ} = 0.

Let S̃ (resp., S) be the class of Specht modules for H̃_{O} (resp., H). Part (a) of the following result is well-known, see, e. g., [DPS2, (7.5)]. Part (b) is proved in [DJ, (7.7)].

Lemma 2.1. (a) For λ ∈ Λ^{+}(r), Ỹ_{λ}, Ỹ_{λ}^{♮} ∈ mod–H̃(S̃). Thus, Y_{λ}, Y_{λ}^{♮} ∈ mod–H(S). In fact, Ỹ_{λ}/S̃_{λ} and Coker(Ỹ_{λ}^{♮} → Ỹ_{λ}) have Specht module filtrations (as do the H-modules obtained from them by base change to k).

(b) For λ ∈ Λ^{+}(r), Y_{λ} is projective if and only if λ ∈ Λ_{e-res}^{+}(r). For λ ∈ Λ_{e-res}^{+}(r), D_{λ′}^{Φ} is isomorphic to both the head and the socle of the projective H-module Y_{λ}.

Duality revisited.

The category of (left or right) H̃_{O}– and H–modules have natural (contravariant) duality functors DH̃_{O} and D_{H} defined using (−)^{∗} = Hom_{O}(−, O) and ι. Also, M̃^{D2 ; H̃_{O}} ≅ M̃ whenever M̃ is a H̃_{O}-module which is O–free of finite rank. Similarly, M^{D_{H}^{2}} ≅ M for all finite dimensional H-modules M. In turn DH̃_{O} (resp., D_{H}) induces a duality D_{S̃_{q}(n,r)} (resp., D_{S_{q}(n,r)}) on S̃_{q}(n, r)-mod (resp., S_{q}(n, r)-mod). See [CPS4, §1.3]. Using the functors (1.0.6), there are commutative diagrams

[figure: S̃_{q}(n,r)−mod^{op}; S̃_{q}(n,r)−mod; _{mod−H̃_{O}^{op}}; ^{mod−H̃_{O}}; (−)^{∘}; (−)^{∘}; DH̃O ,; D_{S̃q(n,r)}; S̃_{q}(n,r)−mod^{op}; S̃_{q}(n,r)−mod; _{mod−H̃_{O}^{op}}; ^{mod−H̃_{O}}; (−)^{∘}; (−)^{∘}; DH̃O; D_{S̃q(n,r)}] (2.1.1)

A similar diagram holds at the field level—in fact, the proofs in [CPS4, (1.2.3.1)] are only given over k, but go over mutatis mutandis to O.

By [DPS2, (7.2c)], the irreducible S_{q}(r, r)-modules are fixed by D_{S_{q}(r,r)}. Lemma 2.1(b) easily implies the same statement holds for irreducible H-modules and D_{H}, since the Y_{λ} are self-dual [DPS2, (1.1)].

Lemma 2.2. Given λ ∈ Λ^{+}(r), let L(λ) be the corresponding irreducible S_{q}(r, r)- module. Then

L(λ)^{⋄} ≅ \begin{cases}D_{λ′}^{Φ}, if λ ∈ Λ_{e-res}^{+}(r) \\ 0, otherwise. \end{cases}

Proof. By [DPS2, (7.3)], S_{λ}^{D_{H}} ≅ S_{λ′}^{Φ}, ∀λ ∈ Λ^{+}(r). Let μ ∈ Λ_{e-res}^{+}(r), so [L(λ)^{⋄} : D_{μ′}^{Φ}] = dim Hom_{H}(Y_{μ}, L(λ)^{⋄}) = dim Hom_{S_{q}(r,r)}(L(λ), Y_{μ}^{⋄}) = dim Hom_{S_{q}(r,r)}(L(λ), P (μ)) by Prop. 1.2 and Lemma 2.1(b). Y_{μ} is projective, so P (μ) = Y_{μ}^{⋄} is a direct summand of T and P (μ) ≅ X(μ) ([DPS2, (5.4)] and Remark 1.1) is self-dual. So dim Hom_{S_{q}(r,r)}(L(λ), P (μ)) = δ_{λ,μ}. □

Remark 2.3. Although we have mostly worked right H̃-modules, the isomorphism (2.0.5) insures that everything holds in essentially the same form for H̃–mod. For example, the definitions of §2.1 imply ι “commutes” with the induction functors (2.0.7) and (2.0.8), and that Ỹ_{λ}^{ι} (resp., S̃_{λ}^{ι}, Ỹ_{λ}^{♮ι}) identifies with the corresponding Young (resp., Specht, twisted Young) indexed by λ in H̃–mod. Then Ỹ_{λ}^{∗} ≅ Ỹ_{λ}^{ι} (resp., S̃_{λ}^{∗} ≅ S̃_{λ′}^{Φι}, Ỹ_{λ}^{♮∗} ≅ Ỹ_{λ}^{♮ι}). We generally denote Ỹ_{λ}^{ι} (resp., S̃_{λ}^{ι}, Ỹ_{λ}^{♮ι}) simply by Ỹ_{λ} (resp., S̃_{λ}, Ỹ_{λ}^{♮}).

An integral Ext^{1}-vanishing result.

For e > 3, the result below for a field can be deduced from [HN, (3.3.4)], and this implies the integral case, although a version of the integral case had already been remarked in [DPS2, (7.4)]. We work with the algebra S̃_{q}(r, r), though the same result holds for any S̃_{q}(n, r) using [CPS6, §2] (a fact we do not need).

Lemma 2.4. Assume that e > 2 in (2.0.17). Let λ ∈ Λ^{+}(r). Then:

(a) Ext1 ; H̃_{O}(Ỹ_{λ}^{♮}, T̃(r, r)) = 0.

(b) Ext1 ; H̃_{O}(Ỹ_{λ}, T̃(r, r)) = 0.

(c) Ext1 ; H̃_{O}(S̃_{λ}, T̃(r, r)) = 0.

Also, S_{λ}^{⋄} ≅ Δ(λ), for all λ ∈ Λ^{+}(r).

Proof. Using (2.0.13), the vanishing results in (a) and (b) come down to showing that if e > 2, then for any J ⊆ S, Ext1 ; H̃_{JO}({ \atop SGN}, ĨND) = Ext1 ; H̃_{JO}(ĨND, ĨND) = 0. However, this is easily checked by direct (matrix) calculation. In particular, (c) holds if λ = (1^{r}) is minimal since then S̃_{λ} ≅ { \atop SGN}. So we prove (c) by upward induction on λ. In what follows, write T̃ for T̃(r, r) and Ã = S̃_{q}(r, r).

In general, we have

(2.4.1) Δ(λ) = Δ̃(λ)_{k} = HomH̃_{O}(S̃_{λ}, T̃)_{k} ↪ Hom_{H}(S_{λ}, T) = S_{λ}^{⋄}.

By [CPS4, (1.5.2)],

S_{λ}^{⋄} ≅ Δ(λ) ⟺ Ext1 ; H̃_{O}(S̃_{λ}, T̃) = 0. (2.4.2)

So by (2.4.1), it suffices to show that Ev_{Δ(λ)} : Δ(λ) → Δ(λ)^{⋄⋄} = S_{λ}^{⋄} is an isomorphism. Form the short exact sequence

0 → Δ̃(λ) → X̃(λ) → C̃(λ) → 0 (2.4.3)

and let C(λ) = C̃(λ)_{k}. Now (−)^{⋄} = Hom_{A}(−, T) is exact since T is a tilting module [DPS2, (5.4)], so base changing to k in (2.4.3) and applying (−)^{⋄} yields an exact sequence 0 → C(λ)^{⋄} → X(λ)^{⋄} → Δ(λ)^{⋄} → 0. This gives a commutative diagram

[figure: 0; Δ(λ); X(λ); C(λ); 0]

[figure: Ev_{Δ(λ)}; Ev_{X(λ)}; Ev_{C(λ)}]

[figure: 0; Δ(λ)^{⋄⋄}; X(λ)^{⋄⋄}; C(λ)^{⋄⋄}]

with exact rows. The module C(λ) has a Δ-filtration with sections Δ(μ), μ ◁ λ, so induction and an easy argument on the filtration length of C(λ) shows that the right most evaluation map Ev_{C(λ)} : C(λ) → C(λ)^{⋄⋄} is an isomorphism. We assert that Ev_{X(λ)} : X(λ) → X(λ)^{⋄⋄} is an isomorphism. By [DPS2, (2.3)], X̃(λ)^{⋄} ≅ Ỹ_{λ}^{♮}, while EvM̃ : M̃ →^{∼} M̃^{⋄⋄} for any H̃-module which is free over O, by the discussion at the end of Remark 1.1. So (a) and [CPS4, (1.5.2)] imply that (X̃(λ)^{⋄⋄})_{k} ≅ ((X̃(λ^{⋄})_{k})^{⋄}. But Lemma 1.4(a) and the fact that T̃ is a tilting module [DPS2, (5.4)] imply that Ext_{S_{q}(r,r)}^{1}(X̃(λ), T̃) = 0, so (X̃(λ)^{⋄})_{k} ≅ X(λ)^{⋄}. Therefore, Ev_{X(λ)} is obtained by base change to k of the isomorphism Ev_{X̃(λ)} : X̃(λ) →^{∼} X̃(λ)^{⋄⋄} [CPS4, (4.15), (4.31)]. Thus, the snake lemma implies that Ev_{Δ(λ)} : Δ(λ) → Δ(λ)^{⋄⋄} is an isomorphism. □

3. Resolutions.

We continue with the notation of subsection 2.1. This section studies a very specific complex associated to any module M̃ for the Hecke algebra H̃. This involves generalizing a construction of Deodhar [Deo] for the trivial module.

3.1. The complex. In this subsection, H̃ can be an arbitrary Hecke algebra over Z corresponding to a finite Coxeter system (W, S). Let |S| = m and, for 0 ≤ i ≤ m, we will consider the functor

(3.0.4) 𝔑_{i} = ∐_{J⊆S,|J|=i} Ind^{⊗}H̃ ; H̃_{J} ∘ ResH̃ ; H̃_{J} : mod–H̃ → mod–H̃.

Fix a linear ordering < on S. If J ⊂ K and J = K \ {s′} for some s′ ∈ S, define ϵ(J, K) = (−1)^{|{s∈J | s<s′}|}. A natural transformation d_{i} : 𝔑_{i} → 𝔑_{i+1} results by setting, for J ⊆ S, |J| = i,

d_{i}|_{Ind_{⊗}H̃ ; H̃_{J} ∘ResH̃ ; H̃_{J}} = ∑_{K⊃J,|K|=i+1} ϵ(J, K)ϕ_{J,K} ∘ Ind^{⊗}H̃ ; H̃_{J} ∘ ResH̃ ; H̃_{J};

cf. (2.0.10). In what follows, Ã ⊗ B̃ denotes Ã ⊗_{Z} B̃, while we abbreviate Ã ⊗H̃_{J} B̃ to Ã ⊗_{J} B̃; if J = {s}, write Ã ⊗_{J} B̃ = Ã ⊗_{s} B̃.

The following result implies that (𝔑_{•}, d_{•}) is a complex of functors.

Lemma 3.1. We have d_{i+1} ∘d_{i} = 0, for 0 ≤ i < m. In particular, (𝔑_{•}, d_{•}) is a complex of functors.

Proof. Let J ⊂ S with |J| + 2 ≤ m. Choose s_{1} < s_{2} in S\J. Let K_{i} = J ⋃{s_{i}} and L = J ⋃{s_{1}, s_{2}}. Then

|{s ∈ J | s < s_{2}}| + |{s ∈ K_{2} | s < s_{1}}| = |{s ∈ J | s < s_{1}}| + |{s ∈ K_{1} | s < s_{2}}| − 1,

so ϵ(K_{1}, L)ϵ(J, K_{1})+ϵ(K_{2}, L)ϵ(J, K_{2}) = 0 and d_{i+1} ∘d_{i} = 0 by transitivity of induction.

□

Lemma 3.2. We have Ker d_{0} ≅ Φ.

Proof. Fix M̃ ∈ mod–H̃, and for simplicity write d_{0} instead of d_{0}(M̃). Given ξ = ∑_{w∈W} m_{w} ⊗ τ_{w} ∈ 𝔑_{0}(M̃),

\begin{aligned}d_{0}(ξ) = 0 & ⟺ 0 = m_{w} ⊗_{s} τ_{w} + m_{sw} ⊗_{s} τ_{sw} = m_{s} ⊗_{s} τ_{w} + m_{sw}τ_{s} ⊗_{s} τ_{w} \\ & ⟺ m_{sw} = −m_{w}τ_{s}^{−1} \end{aligned}

for all w ∈ W, s ∈ S with sw > w. Thus, there is a bijection

π : M̃ ⟶^{∼} Ker(d_{0}), m ↦ ∑_{w} ϵ_{w}mτ_{w}^{−1} ⊗ τ_{w}

of Z-modules. Here ϵ_{w} = (−1)^{ℓ(w)}.

16 BRIAN J. PARSHALL AND LEONARD L. SCOTT

To verify that π : M̃^{Φ} →^{∼} Ker(d_{0}) in mod–H̃, we check that π(mΦ(τ_{s})) = π(m)τ_{s}, ∀m ∈ M̃, s ∈ S. (3.2.1) Let w ∈ W, s ∈ S with ws > w. Since Φ(τ_{s}) = −tτ_{s}^{−1},

\begin{matrix}ϵ_{w}mΦ(τ_{s})τ_{w}^{−1} ⊗ τ_{w} + ϵ_{ws}mΦ(τ_{s})τ_{ws}^{−1} ⊗ τ_{ws} \\ = −tϵ_{w}mτ_{s}^{−1}τ_{w}^{−1} ⊗ τ_{w} − tϵ_{ws}mτ_{s}^{−2}τ_{w}^{−1} ⊗ τ_{ws}. \end{matrix}

But −tτ_{s}^{−2} = −1 + (t − 1)τ_{s}^{−1}, so this expression equals tϵ_{ws}mτ_{ws}^{−1} ⊗ τ_{w} + ϵ_{ws}m(−1 + (t − 1)τ_{s}^{−1})τ_{w}^{−1} ⊗ τ_{ws}

= ϵ_{ws}mτ_{ws}^{−1} ⊗ (tτ_{w} + (t − 1)τ_{ws}) + ϵ_{w}mτ_{w}^{−1} ⊗ τ_{w}τ_{s}
= (ϵ_{ws}mτ_{ws}^{−1} ⊗ τ_{ws})τ_{s} + (ϵ_{w}mτ_{w}^{−1} ⊗ τ_{w})τ_{s}

which immediately implies (3.2.1). □

The following result also plays an important role in the next subsection. Theorem 3.3. Let L ⊆ S. The complex 𝔑_{•} (Ind^{⊗}H̃ ; H̃_{L} ĨND) is isomorphic to the complex (3.3.1) Ind^{⊗}H̃ ; H̃_{L} ∘ ResH̃ ; H̃_{L} ( 𝔑_{•} (ĨND)). Proof. Let J, L ⊆ S. We define a right H̃-module map

f_{J,L} : Ind^{⊗}H̃ ; H̃_{J} (ResH̃ ; H̃_{J} ∘Ind^{⊗}H̃ ; H̃_{L} ĨND) → Ind^{⊗}H̃ ; H̃_{L}(ResH̃ ; H̃_{J} ∘Ind^{⊗}H̃ ; H̃_{J} ĨND)

by the formulas 1 ⊗_{L} τ_{b} ⊗_{J} h ↦ t^{−ℓ(b)}1 ⊗_{J} τ_{b^{−1}} ⊗_{L} τ_{b}h, b ∈ D_{L}, h ∈ H̃. (3.3.2) We must show that f_{J,L} is well-defined. (It will then automatically be an H̃-module map.) As a Z-module,

Ind^{⊗}H̃ ; H̃_{J} (ResH̃ ; H̃_{J} ∘Ind^{⊗}H̃ ; H̃_{L} ĨND) = Z ⊗_{L} H̃ ⊗_{J} H̃

= ⨁_{d∈D_{L,J} ,v∈D_{L^{d}∩J}} Z ⊗_{L} τ_{d} ⊗_{J} τ_{v}

Thus, we can define a Z-linear map

f′_{J,L} : Ind^{⊗}H̃ ; H̃_{J} (ResH̃ ; H̃_{J} ∘Ind^{⊗}H̃ ; H̃_{L} ĨND) → Ind^{⊗}H̃ ; H̃_{L}(ResH̃ ; H̃_{L} ∘Ind^{⊗}H̃ ; H̃_{J} ĨND)

by putting

f′_{J,L}(1 ⊗_{L} τ_{d} ⊗_{J} τ_{v}) = t^{−ℓ(d)}1 ⊗_{J} τ_{d^{−1}} ⊗_{L} τ_{d}τ_{v}, d ∈ D_{L,J}, v ∈ D_{L^{d}∩J}.

We will show that f′_{J,L} = f_{J,L}. \underline{Claim.} f′_{J,L}(1⊗_{L}τ_{d}⊗_{J} τ_{z}τ_{v}) = t^{−ℓ(d)}1⊗τ_{d^{−1}}⊗_{L}τ_{d}τ_{z}τ_{v}, whenever d ∈ D_{L,J}, z ∈ W_{L^{d}∩J}, and v ∈ D_{L^{d}∩J}. To see this, note that ℓ(dz) = ℓ(d)+ℓ(z) and ℓ(dzd^{−1}d) = ℓ(dzd^{−1})+ℓ(d). In particular, ℓ(dzd^{−1}) = ℓ(z). Thus, 1 ⊗_{L} τ_{d} ⊗_{J} τ_{z}τ_{v} = 1 ⊗_{L} τ_{d}τ_{z} ⊗_{J} τ_{v} = 1 ⊗_{L} τ_{dzd^{−1}}τ_{d} ⊗_{J} τ_{v} = t^{ℓ(z)}1 ⊗_{L} τ_{d} ⊗_{J} τ_{v}. Similarly, 1 ⊗_{J} τ_{d^{−1}} ⊗_{L} τ_{d}τ_{z}τ_{v} = 1 ⊗_{J} τ_{d^{−1}} ⊗_{L} τ_{dzd^{−1}}τ_{d}τ_{v} = 1 ⊗_{J} τ_{d^{−1}}τ_{dzd^{−1}} ⊗_{L} τ_{d}τ_{v} = 1 ⊗_{J} τ_{zd^{−1}} ⊗_{L} τ_{d}τ_{v} = 1 ⊗_{J} τ_{z}τ_{d^{−1}} ⊗_{L} τ_{d}τ_{v}, since d^{−1} ∈ D_{J,L}. The last expression is t^{ℓ(z)}1 ⊗ τ_{d^{−1}} ⊗ τ_{d}τ_{v}. Using these facts, we have

f′_{J,L}(1 ⊗_{L} τ_{d} ⊗_{J} τ_{z}τ_{v}) = f′_{J,L}(t^{ℓ(z)}1 ⊗_{L} τ_{d} ⊗_{J} τ_{v})
= t^{ℓ(z)}t^{−ℓ(d)}1 ⊗_{J} ⊗_{d^{−1}} ⊗_{L} τ_{d}τ_{v}
= t^{−ℓ(d)}1 ⊗_{J} τ_{d^{−1}} ⊗_{L} τ_{d}τ_{z}τ_{v},

which proves the claim.

Now let b ∈ D_{L}, h ∈ H̃ be arbitrary. Write b = dy where d ∈ D_{L,J} and y ∈ W_{J}. (Certainly, b = udy, where u ∈ W_{L}, d ∈ D_{L,J}, y ∈ W_{J} with ℓ(b) = ℓ(u) + ℓ(y) + ℓ(y) by Howlett’s theorem; see also [CR, (64.38)]. However, b ∈ D_{L}, so u = 1.) Express the product τ_{y}h as a linear combination τ_{y}h = ∑_{z∈W_{L^{d}∩J} ,v∈D_{L^{d}∩J}} c_{z,v}τ_{z}τ_{v}, where c_{z,v} ∈ Z. Then

f′_{J,L}(1 ⊗_{L} τ_{b} ⊗_{J} h) = f′_{J,L}(1 ⊗_{L} τ_{d}τ_{y} ⊗_{J} h)
= f′_{J,L}(1 ⊗_{L} τ_{d} ⊗_{J} τ_{y}h)
= f′_{J,L}(∑ c_{z,v}1 ⊗_{L} τ_{d} ⊗_{J} τ_{z}τ_{v})
= ∑ c_{z,v}t^{−ℓ(d)}1 ⊗_{J} τ_{d^{−1}} ⊗_{L} τ_{d}τ_{z}τ_{v}
= t^{−ℓ(d)}1 ⊗_{J} τ_{d^{−1}} ⊗_{L} τ_{d}τ_{y}h
= t^{−ℓ(d)−ℓ(y)}1 ⊗_{J} τ_{y^{−1}}τ_{d^{−1}} ⊗_{L} τ_{d}τ_{y}h
= t^{−ℓ(b)}1 ⊗_{J} τ_{b^{−1}} ⊗_{L} τ_{b}h,

which is our proposed definition of f_{J,L}(1 ⊗_{L} τ_{b} ⊗_{J} h). Thus, f_{J,L} is well-defined. For J ⊆ K, (3.3.2) immediately implies there is a commutative diagram

Ind^{⊗}H̃ ; H̃_{J} (ResH̃ ; H̃_{J} ∘Ind^{⊗}H̃ ; H̃_{L} ĨND)  \overline{ϕ_{J,K}}  \overline{}Ind^{⊗}H̃ ; H̃_{K}(ResH̃ ; H̃_{K} ∘Ind^{⊗}H̃ ; H̃_{L} ĨND)
[figure: f_{J,L}; f_{K,L}]
Ind^{⊗}H̃ ; H̃_{L}(ResH̃ ; H̃_{L} ∘Ind^{⊗}H̃ ; H̃_{J} ĨND) \overline{Ind^{⊗}H̃ ; H̃_{L}∘ResH̃ ; H̃_{L} ∘ϕ_{J,K}} \overline{}Ind^{⊗}H̃ ; H̃_{L}(ResH̃ ; H̃_{L} ∘Ind^{⊗}H̃ ; H̃_{K} ĨND).

The lemma follows directly from the definition of the complex 𝔑_{•} of functors. □

For M̃ ∈ mod–H̃, we denote the complex

0 → M̃^{Φ} → 𝔑_{0}(M̃) → ⋯ → 𝔑_{m}(M̃) → 0

by

0 → 𝔑′_{−1}(M̃) → 𝔑′_{0}(M̃) → ⋯ → 𝔑′_{m}(M̃) → 0. (3.3.3)

In the special case when the complex (𝔑_{•}, d_{•}) is evaluated at M̃ = ĨND, it has been proved by Deodhar [Deo] that one obtains a resolution of { \atop SGN}, i. e., the complex 𝔑′_{•}(M̃) in (3.3.3) is acyclic when M̃ = ĨND. We will use that result in proving that (3.3.3) is acyclic for any finitely generated H̃-module M̃.^{3}

Theorem 3.4. The complex (𝔑_{•}, d_{•}) of functors on mod–H̃ has cohomology given by

H^{i}(𝔑_{•}) = \begin{cases}Φ, i = 0 \\ 0, i > 0. \end{cases}

Proof. We must show for any M̃ ∈ mod–H̃ that the complex (𝔑′_{•}(M̃), d_{•}) in (3.3.3) is acyclic. By (3.2), H^{i}(𝔑′_{•}(M̃)) = 0 for all M̃ if i = −1, 0.

Note that the functors Ind^{⊗}H̃ ; H̃_{J} ∘ResH̃ ; H̃_{J}, J ⊆ S and the functor Φ are exact on mod–H̃. Thus, if 0 → X̃_{1} → X̃_{2} → X̃_{3} → 0 is a short exact sequence in mod–H̃,

0 → 𝔑′_{•}(X̃_{1}) → 𝔑′_{•}(X̃_{2}) → 𝔑′_{•}(X̃_{3}) → 0

is a short exact sequence complexes in mod–H̃. If 𝔑′_{•}(X_{i}) is acyclic for any two of {1, 2, 3}, then these complexes are acyclic for all i = 1, 2, 3 by the long exact sequence of cohomology. In particular, if X̃ ∈ mod–H̃ has a finite filtration with sections F̃_{t} such that for each t, 𝔑′_{•}(F̃_{t}) is acyclic, then it follows by induction on the filtration length that 𝔑′_{•}(X̃) is acyclic.

Given M̃ ∈ mod–H̃, the right H̃-module M̃ ⊗_{Z} H has a filtration with sections of the form Ind^{⊗}H̃ ; H̃_{∅}(ĨND_{Z′}), where ĨND_{Z′} = Z′ ⊗_{Z} ĨND, for various rings Z′ which are homomorphic images of Z. This follows because, as a Z-module, M̃ is finitely generated. The complex 𝔑′_{•}(Ind^{⊗}H̃ ; H̃_{∅} ĨND) identifies, using Thm. 3.3 with L = ∅, to the complex 𝔑′_{•}(ĨND) ⊗_{Z} H̃ which is acyclic by [Deo, (5.1)]. As an exact bounded complex of free Z-modules, it remains exact after base change to Z′, i. e., the complex 𝔑′_{•}(Ind^{⊗}H̃ ; H̃_{∅} ĨND_{Z′}) is acyclic. Thus, by the previous paragraph, 𝔑′_{•}(M̃⊗_{Z} H̃) is acyclic.

Now, given M̃ ∈ mod–H̃, form the exact sequence 0 → M̃′ → M̃⊗_{Z} H̃ ⟶^{mult} M̃ → 0. Write Ỹ = M̃⊗_{Z} H̃, and form the exact sequence 0 → 𝔑′_{•}(M̃′) → 𝔑′_{•}(Ỹ) → 𝔑′_{•}(M̃) → 0. By above, the middle complex 𝔑′_{•}(Ỹ) is acyclic. Obviously, H^{m}(𝔑′_{•}(M̃)) = 0 for any M̃, so, by the long exact sequence of cohomology, H^{m−1}(𝔑′_{•}(M̃)) = 0 provided m > 1. We repeat this argument if M̃ is replaced by M̃′ to conclude that H^{m−1}(𝔑′_{•}(M̃′)) = 0. Thus, if m > 2, H^{m−2}(𝔑′(M̃)) = 0. Continuing gives H^{i}(𝔑′_{•}(M̃)) = 0 for all i. □

Remarks 3.5. (a) (Base change to k) Suppose that M̃ ∈ mod–H̃ is a Z-projective and put M = M̃_{k} for some commutative Z-algebra k. Then 0 → M̃^{Φ} → 𝔑_{•}(M̃) → 0 is a bounded acyclic complex of projective Z-modules, so it remains acyclic after base change to k. In fact, k ⊗ 𝔑_{•}(M̃) ≅ 𝔑_{•}(M).

3It follows from the proof that the theorem holds for arbitrary H̃-modules.

(b) S. Schroll has pointed out to us a result on bimodule complexes in her unpublished thesis, [Sch, Thm. 2.4.1], which can be shown equivalent to Thm. 3.4. Her proof is different, however. In particular, she relies on acyclic filtrations and establishes no analogue of Thm. 3.3, which plays an essential role in our proof and applications below, cf. Thm. 3.7.

3.2. Passing to the q-Schur algebra. We now specialize to W = 𝔖_{r}. We consistently work over a local system (O, K, k) satisfying the conditions set down in the second paragraph of Subsection 2.2. We also assume that e > 2 in (2.0.17). Recall S̃_{q}(r, r) = EndH̃_{O}(T̃), where T̃ := T̃(r, r)_{O}.

We first establish some preliminary results on Specht filtrations.

Lemma 3.6. Assume e > 2. Let Ñ_{i} = 𝔑_{i}(ĨND) ∈ mod–H̃_{O}. In the exact sequence,

0 → { \atop SGN} → Ñ_{0} → ⋯ → Ñ_{m} → 0 (3.6.1)

obtained by taking M̃ = ĨND ∈ mod–H̃_{O} in Thm. 3.4 (with m = r − 1), the kernels K̃_{i} = Ker(Ñ_{i} → Ñ_{i+1}) ∈ mod–H̃_{O} have Specht filtrations.

Proof. By (2.3), it suffices to prove the result by replacing mod–H̃_{O} by the category H̃_{O}–mod and Thm. 3.4 by its corresponding left module analogue. This will make it possible to quote the results of [CPS5]. In this proof, denote S̃_{q}(r, r) simply by Ã.

First, each Ñ_{i}^{∗} = Hom_{O}(Ñ_{i}, O) ∈ mod–H̃_{O} is a direct sum of modules T̃_{λ}, where |J(λ)| = i. Thus, Ñ_{i}^{∗Φ⋄} is a tilting module for Ã [DPS2, (5.4)]; hence, Ñ_{i}^{∗Φ⋄} has both Δ̃– and ˜∇–filtrations.

Now we can apply [CPS5, (11.10)] to obtain, for each i, an exact sequence

0 → K̃_{i}^{∗Φ⋄} → Ñ_{i}^{∗Φ⋄} → K̃_{i+1}^{∗Φ⋄} → 0 (3.6.2)

in Ã–mod. When i = m − 1, we obtain the exact sequence

0 → K̃_{m−1}^{∗Φ⋄} → Ñ_{m−1}^{∗Φ⋄} → Ñ_{m}^{∗Φ⋄} → 0. (3.6.3)

Since both Ñ_{m−1}^{∗Φ⋄} and Ñ_{m}^{∗Φ⋄} have Δ̃-filtrations, Lemma 1.4(b) implies that K̃_{m−1}^{∗Φ⋄} ∈ Ã– mod(Δ̃). Working backwards, we conclude that each K̃_{i}^{∗Φ⋄} has a Δ̃-filtration.

For each i, the discussion at the end of Remark 1.1 implies that EvK̃_{i}^{∗Φ} : K̃_{i}^{∗Φ} → K̃_{i}^{∗Φ⋄⋄}

is an isomorphism. Since K̃_{i}^{∗Φ⋄} has a Δ̃-filtration and since (−)^{⋄} = HomÃ(−, T̃) is exact, it follows that K̃_{i}^{∗Φ⋄⋄} has a Specht filtration. Thus, K̃_{i}^{∗Φ} has a Specht filtration. Since (−)^{∗Φ} takes right Specht modules to left Specht modules, so each K̃_{i} has a Specht filtration. □

We give two results which determine how the resolution 𝔑(X̃) behaves under the diamond operation HomH̃(−, T̃). First, we consider the case of Young modules.

20 BRIAN J. PARSHALL AND LEONARD L. SCOTT

Theorem 3.7. Assume e > 2. For μ ∈ Λ^{+}(r), applying (−)^{⋄} to the resolution

0 → Ỹ_{μ}^{Φ} → 𝔑_{0}(Ỹ_{μ}) → ⋯ → 𝔑_{m}(Ỹ_{μ}) → 0

yields a projective resolution 0 → 𝔑_{m}(Ỹ_{μ})^{⋄} → ⋯ → 𝔑_{0}(Ỹ_{μ})^{⋄} → Ỹ_{μ}^{Φ⋄} → 0 (3.7.1) of X(μ′) = Ỹ_{μ}^{Φ⋄} in S̃_{q}(r, r)–mod. Proof. Since Ỹ_{μ} is a direct summand of T̃_{μ} = Ind^{⊗}H̃_{O} ; H̃_{λO} ĨND, it suffices to show 0 → T̃_{μ}^{Φ} → 𝔑_{0}(T̃_{μ}) → ⋯ → 𝔑_{m}(T̃_{μ}) → 0 (3.7.2) remains exact after applying the functor (−)^{⋄} = HomH̃_{O}(−, T̃). But, using Thm. 3.3 and the fact that the functor Φ commutes with the induction functors, we can replace (3.7.2) by the exact complex

0 → Ind^{⊗}H̃_{O} ; H̃_{μO}(ĨND^{Φ}) → Ind^{⊗}H̃_{O} ; H̃_{O}μ ∘ ResH̃_{O} ; H̃_{μO}(𝔑_{0}ĨND) → ⋯
⋯ → Ind^{⊗}H̃_{O} ; H̃_{μO} ∘ ResH̃_{O} ; H̃_{μO}(𝔑_{𝔪}ĨND) → 0

(3.7.3)

and show that this complex remains exact after applying (−)^{⋄}.

Since Ind^{⊗}H̃_{O} ; H̃_{μO} is an exact functor,

Ker(Ind^{⊗}H̃_{O} ; H̃_{μO}∘IndH̃_{O} ; H̃_{μO}(𝔑_{i}ĨND) → Ind^{⊗}H̃_{O} ; H̃_{μO}∘ResH̃_{O} ; H̃_{μO}(𝔑_{i+1}ĨND) ≅ Ind^{⊗}H̃_{O} ; H̃_{μO}∘ResH̃_{O} ; H̃_{μO}(K̃_{i}),

in the notation of Lemma 3.6. Now by the Eckmann-Shapiro Lemma,

Ext1 ; H̃_{O}(Ind^{⊗}H̃_{O} ; H̃_{μO} ∘ ResH̃_{O} ; H̃_{μO} K̃_{i}, T̃) ≅ Ext1 ; H̃_{μO}(ResH̃_{O} ; H̃_{μO} K̃_{i}, ResH̃_{O} ; H̃_{μO} ĨND).

By (2.0.13), ResH̃_{O} ; H̃_{μO} T̃ is a direct sum of H̃_{μO}-modules of the form

Ind^{⊗}H̃_{μO} ; H̃_{JO}(ResH̃_{O} ; H̃_{JO} ĨND),

where J runs over subsets of J(μ). To show that (3.7.3) remains exact after applying (−)^{⋄} it suffices to show that the above Ext1 ; H̃_{O} equals 0. Thus, it suffices to show that Ext1 ; H̃_{μO}(ResH̃_{O} ; H̃_{μO} K̃_{i}, Ind^{⊗}H̃_{μO} ; H̃_{JO} ĨND) = 0. The Echmann-Shapiro Lemma, this time for the right adjoint IndH̃_{O} ; H̃_{μO} of ResH̃_{O} ; H̃_{μO}, yields

Ext1 ; H̃_{μO}(ResH̃_{O} ; H̃_{μO} K̃_{i}, Ind^{⊗}H̃_{μO} ; H̃_{JO} ĨND) ≅ Ext1 ; H̃_{O}(K̃_{i}, IndH̃_{O} ; H̃_{μO} Ind^{⊗}H̃_{μO} ; H̃_{JO} ĨND)

≅ Ext1 ; H̃_{O}(K̃_{i}, Ind^{⊗}H̃_{O} ; H̃_{JO} ĨND) = 0,

by (2.0.12), Lemma 3.6 and Lemma 2.4(c). □

QUANTUM WEYL RECIPROCITY FOR COHOMOLOGY 21

Now we consider the more difficult case of Specht modules. Theorem 3.8. Assume e > 2. For μ ∈ Λ^{+}(r), consider the complex of S̃_{q}(r, r)-modules 0 ← 𝔑_{0}(S̃_{μ})^{⋄} ← ⋯ ← 𝔑_{m}(S̃_{μ})^{⋄} ← 0 (3.8.1) obtained by applying (−)^{⋄} to the complex 𝔑_{•}(S̃_{μ}). It has trivial cohomology in positive degrees and cohomology ˜∇(μ′) in degree 0 and thus provides a resolution 0 → 𝔑_{m}(S̃_{μ})^{⋄} → ⋯ → 𝔑_{0}(S̃_{μ})^{⋄} → ˜∇(μ′) → 0. (3.8.2) Proof. Write Ã = S̃_{q}(r, r). By Thm. 3.7, 𝔑_{•}(Ỹ_{μ})^{⋄} → Ỹ_{μ}^{Φ⋄} → 0 is a resolution. \underline{Step 1.} We construct a special resolution of S̃_{μ}^{Φ} by modules of the form ⨁ Ỹ_{ν}^{Φ}: Given ω ∈ Λ^{+}(r), Lemma 1.5(b) provides a resolution 0 → Δ̃(ω) → X̃_{0} → ⋯ → X̃_{n} → 0, (3.8.3) where the X̃_{i} are tilting modules and, for 0 ≤ i < n, Ker(X̃_{i} → X̃_{i+1}) has a Δ̃-filtration. For any M̃ ∈ Ã–mod, (2.1.1) states that M̃^{DÃ(−)⋄} ≅ M̃^{⋄DH̃_{O}}.

Applying (−)^{⋄} to (3.8.3) gives an exact sequence as does applying the duality DH̃_{O} to the result.^{4} Therefore, (−)^{DÃ⋄} also gives an exact sequence, namely, 0 → ˜∇(ω)^{⋄} → X̃_{0}^{⋄} → ⋯ → X̃_{n}^{⋄} → 0, (3.8.4) since the X̃_{i} ≅ X̃_{i}^{DÃ}. Also, ˜∇(ω)^{⋄} ≅ S̃_{ω′}^{Φ}, by (2.0.16) and X̃_{i}^{⋄} = ⨁_{j} Ỹ_{ν_{ij}}^{Φ}. Taking ω′ = μ, we can rewrite (3.8.4) as an exact complex 0 → S̃_{μ}^{Φ} → X̃_{0}^{⋄} → ⋯ → X̃_{n}^{⋄} → 0, (3.8.5) with each X̃_{i}^{⋄} ≅ ⨁_{j} Ỹ_{ν_{ij}}^{Φ}. This completes Step 1. \underline{Step 2.} By construction, all kernels and cokernels in the complex (3.8.3) have a Δ̃- filtration. Thus, when the composite functor (−)^{⋄DH̃_{O}} is applied to obtain (3.8.5), all kernels and cokernels in that complex are filtered by dual Specht modules S̃_{τ}^{DH̃_{O}} ≅ S̃_{τ′}^{Φ}. \underline{Step 3.} By [CPS4, (4.3.1)], EvX̃_{i} : X̃_{i} →^{∼} X̃_{i}^{⋄⋄} is an isomorphism. Thus, applying (−)^{⋄} to the complex 0 → X̃_{0}^{⋄} → ⋯ → X̃_{n}^{⋄} → 0 obtained from (3.8.5), we obtain the complex

0 ← X̃_{0}^{DÃ} ← ⋯ ← X̃_{n}^{DÃ} ← 0

obtained by applying DÃ in (3.8.3). It is thus exact except in degree 0, where its cohomology is Δ̃(μ′)^{DÃ} ≅ ˜∇(μ′).

4Notice that DH̃_{O} takes acyclic complexes of H̃_{O}-modules which are free over O to acyclic complexes.

\underline{Step 4.} Now we construct a double complex Z = \{Z_{i,j}\}_{i,j≥0} by taking Z_{i,j} = 𝔑_{i}(X̃_{j}^{⋄Φ})^{⋄}. The “horizontal” differential d_{i+1}^{hor} : Z_{i+1,j} → Z_{i,j} is defined from Thm. 3.4 to be

d_{i+1,j}^{hor} = (−1)^{i}d_{i}(X_{j}^{⋄Φ})^{⋄}.

If ˜∂_{j} : X̃_{j}^{⋄} → X̃_{j+1}^{⋄} is the differential in (3.8.5), the vertical differential d_{i,j+1}^{ver} : Z_{i,j+1} → Z_{i,j} is defined to be

d_{i,j+1}^{ver} = 𝔑_{i}(˜∂_{j}^{Φ})^{⋄}.

We compute the homology H_{•}(Tot(Z)) of the double complex Tot(Z) in two ways. First, we use the spectral sequence

IIE_{p,q}^{2} = H_{q}^{ver}H_{p}^{hor}(Z) ⟹ H_{p+q}(Tot(Z)).

But H_{p}^{hor}(Z) ≅ X̃_{p}^{⋄⋄} by Thm. 3.7, so by Step 3, ^{II}E_{•,•}^{2} ≅ ^{II}E_{0,0}^{2} ≅ ˜∇(μ) and so IIE_{•,•}^{2} ≅ ^{II}E_{•,•}^{∞}. Second, we use the spectral sequence

IE_{p,q}^{2} = H_{p}^{hor}H_{q}^{ver}(Z) ⟹ H_{p+q}(Tot(Z)).

By Step 2, Lemma 2.4 and the Eckmann-Shapiro Lemma,^{I}E_{•,•}^{2} = ^{I}E_{p,0}^{2} = H_{p}(𝔑_{•}(S̃_{μ})^{⋄}). We also have ^{I}E_{•,•}^{∞} = ^{I}E_{•,•}^{2}. Comparison of the two answers proves the theorem. □

Let Ĩ(μ) = P (μ)^{DÃ} where Ã = S̃_{q}(r, r). Then Ĩ(μ)^{⋄} ≅ P̃(μ)^{DÃ⋄} ≅ P̃(μ)^{⋄DH̃} ≅ Ỹ(μ)^{DH̃} ≅ Ỹ_{μ}. By analogy with (3.8.3), we can choose a tilting resolution of P̃(μ), and repeat the previous argument. We obtain part (a) of the following corollary to the proof. Part (b) is easily proved by induction on the filtration length, using Lemma 2.4.

Corollary 3.9. Suppose that e > 2. Then:

(a) For μ ∈ Λ^{+}(r), there is a resolution

0 → 𝔑_{m}(Ỹ_{μ}^{Φ})^{⋄} → ⋯ → 𝔑_{0}(Ỹ_{μ}^{Φ})^{⋄} → Ĩ(μ) → 0.

(b) If a right H̃_{O}-module Ỹ has a Specht filtration, then the complex 𝔑_{•}(Ỹ)^{⋄} is acyclic except possibly in degree 0, where H^{0}(𝔑_{•}(Ỹ)) has a ˜∇-filtration.

Remark 3.10. The complexes 𝔑_{•}(Ỹ_{λ})^{⋄}, 𝔑_{•}(S̃_{λ})^{⋄} and 𝔑_{•}(Ỹ_{λ}^{Φ})^{⋄} considered in this subsection provide resolution which are free over O and hence remain resolutions upon passage to k. It is always true that 𝔑_{•}(Ỹ_{λ})_{k}^{⋄} ≅ 𝔑_{•}(Y_{λ})^{⋄} by Lemma 2.4. Let Ỹ be any H̃_{O}- mdoule which is O-free. In case i ≤ e − 2, H_{J} is semisimple for any J ⊆ S with |J| = i since ℘_{W_{J}} (t̄) ≠ 0 from the definition of e (2.0.17). Hence, Ext1 ; H̃_{JO}(𝔑_{i}(Ỹ), T̃) = 0, so

that 𝔑_{i}(Ỹ)_{k}^{⋄} ≅ 𝔑_{i}(Y)^{⋄} when 0 ≤ i ≤ e − 2. For any i, Ext1 ; H̃_{O}(𝔑_{i}(S̃_{λ}), T̃) is by the Eckmann-Shapiro Lemma and (2.0.13) isomorphic to a direct sum of various groups Ext1 ; H̃_{O}(S̃_{λ}, T̃_{μ}). Hence, by Lemma 2.4, Ext1 ; H̃_{O}(𝔑_{i}(S̃_{λ}), T̃) = 0, so 𝔑_{•}(S̃_{λ})_{k}^{⋄} ≅ 𝔑_{•}(S_{λ})^{⋄}.

4. Cohomology, I: isomorphisms.

We maintain the notation of Subsection 2.2. Let T̃ = T̃(r, r) and T = T̃_{k}.

Lemma 4.1. Assume that e > 2. For λ ∈ Λ^{+}(r) and 0 ≤ i ≤ e − 2,

Ext_{S_{q}(r,r)}^{i}(∇(λ), S_{q}(r, r)) ≅ Ext_{H}^{i}(S_{λ}, T).

Proof. By Remark 3.10 and Thm. 3.8, there is a resolution

0 → 𝔑_{m}(S_{λ})^{⋄} → ⋯ → 𝔑_{0}(S_{λ})^{⋄} → ∇(λ′) → 0.

Also, 𝔑_{i}(S_{λ}) is both projective and injective for i ≤ e − 2.

Since T is an injective S_{q}(r, r)-module, Cor. 1.3 implies there is a spectral sequence

(4.1.1) E_{2}^{p,q} = Ext_{S_{q}(r,r)}^{p}(∇(λ), Ext_{H}^{q}(T, T)) ⟹ Ext_{H}^{p+q}(T, ∇(λ)^{⋄}).

By the Eckmann-Shapiro Lemma and (2.0.13), for any i, Ext_{H}^{i}(T, T) is a direct sum of terms Ext_{H_{J}}^{i} (IND, IND), J ⊆ S. To compute these groups, we use Thm. 3.4 at M̃ = IND. But Res_{H_{J}}^{H} 𝔑_{i}(SGN) is injective for i ≤ e − 2, and Hom_{H_{J}} (IND, 𝔑_{i}(SGN)) = 0 for all i > 0 by Frobenius reciprocity. Thus, Ext_{H}^{i}(T, T) = 0 for 0 < i ≤ e − 2. Since ∇(λ)^{⋄} ≅ S_{λ′}^{Φ} (2.0.16), (4.1.1) implies that

Ext_{S_{q}(r,r)}^{i}(∇(λ), S_{q}(r, r)) ≅ Ext_{H}^{i}(T, S_{λ′}^{Φ}), 0 ≤ i ≤ e − 2.

Since T is self-dual and S_{λ′}^{Φ} ≅ S_{λ}^{D_{H}}, the proof is complete. □

Theorem 4.2. Assume that e > 2. Let λ, μ ∈ Λ^{+}(r), but assume that μ is not erestricted. Then for 0 ≤ i ≤ e − 2, we have

Ext_{S̃_{q}(r,r)}^{i}( ˜∇(λ), L(μ)) ≅ Ext_{S_{q}(r,r)}^{i}(∇(λ), L(μ)) = 0.

Proof. Suppose that 0 ≤ i ≤ e−2 and Ext_{S_{q}(r,r)}^{i}(∇(λ), L(μ)) ≠ 0. Then P (μ) is a direct summand of the projective module 𝔑_{i}(S_{λ})^{⋄} = 𝔑_{i}(S̃_{λ})_{k}^{⋄}. Thus, Y_{μ} is a direct summand of the projective module 𝔑_{i}(S_{λ}). By Lemma 2.1(b), μ ∈ Λ_{e-res}^{+}(r), as required. □

Remarks 4.3. (a) As with many results in this paper, the above theorem and lemma can be strengthened to statements involving S_{q}(n, r) for all positive integers n ≥ r, using the discussion in the first paragraph of §6.

(b) When q = 1 and i = 0, Thm. 4.2 above specializes to a well-known result of James [J], which states that, given λ ∈ Λ^{+}(r, r), the socle of Δ(λ) consists of irreducible S(r, r)-modules with highest weights which are restricted. The assumption that e > 2 can be removed. See the remark following the proof of Cor. 5.3 below. For other arguments in the i = 0 case and q arbitrary, see [D3, §3.4] and [HN, (3.4.1)].

Lemma 4.4. Assume that e > 2 and let λ ∈ Λ^{+}(r). Then for 0 < i ≤ e − 3, we have Ext_{S_{q}(r,r)}^{i}(∇(λ), S_{q}(r, r)) = 0.

Proof. Write A = S_{q}(r, r) and let B = A^{D_{A}}. By (2.1.1) and the self-duality of tilting modules, B^{⋄} = A^{D_{A}⋄} = A^{⋄D_{H}} = T^{D_{H}} = T. By [CPS4, (1.1.6.3)], applying (−)^{⋄} = Hom_{A}(−, T) to the evaluation map Ev_{B} : B → B^{⋄⋄} yields a surjection B^{⋄⋄⋄} ↠ B^{⋄}. Hence, Ev_{B}^{⋄} : B^{⋄⋄⋄} → B^{⋄} is an isomorphism because both B^{⋄} and B^{⋄⋄⋄} identify with T. Since B^{⋄⋄} ≅ A and (−)^{⋄} = Hom_{A}(−, T) is an exact functor, we obtain an exact sequence 0 → K → B → A → C → 0, where K = Ker(B → B^{⋄⋄} = A) and C = Coker(B → B^{⋄⋄} = A) such that K^{⋄} = C^{⋄} = 0. Thus, the composition factors of K, C are of the form L(μ), where μ ∉ Λ_{e-res}^{+}(r) by Lemma 2.2. Because B is an injective A-module, Ext_{A}^{i}(∇(λ), B) = 0 for all i > 0. Hence, the result follows from the long exact sequence of cohomology plus Thm. 4.2. □

As a consequence of Lemmas 4.4 and 4.1, we obtain:

Theorem 4.5. Assume that e > 2. Then for any λ ∈ Λ^{+}(r), we have

Ext_{H}^{i}(S_{λ}, T) = 0, 0 < i ≤ e − 3.

We now obtain generalizations to Hecke algebras of the results involving cohomology of symmetric groups given in [KN]. We state our results in the same format as [KN, (6.2) & (6.4)]. For the reader’s convenience, we use both the ‘diamond’ functor (−)^{⋄} = Hom_{S_{q}(r,r)}(−, T) : S_{q}(r, r)–mod → mod–H (1.0.6) and the Schur functor F = (−)^{⋄∗} : S_{q}(r, r)–mod → H–mod (1.1.1). Thus, statement (a) in the theorem below involves both the right H-module S_{λ} and the left H-module S_{λ′}^{Φ}—cf. Remark 2.3. In statement (b), recall our convention: If λ ∉ Λ_{e-reg}^{+}(r), we set D_{λ} = 0.

Theorem 4.6. Assume that e > 2 and that 0 ≤ i ≤ e − 3. Let M ∈ S_{q}(r, r)–mod.

(a) Ext_{S_{q}(r,r)}^{i}(M, Δ(λ)) ≅ Ext_{H}^{i}(S_{λ}, M^{⋄}) ≅ Ext_{H}^{i}(FM, S_{λ′}^{Φ}). Also, there is an exact sequence

0 → Ext_{S_{q}(r,r)}^{e−2}(M, Δ(λ)) → Ext_{H}^{e−2}(S_{λ}, M^{⋄}) → Hom_{S_{q}(r,r)}(M, Ext_{H}^{e−2}(S_{λ}, T))
→ Ext_{S_{q}(r,r)}^{e−1}(M, Δ(λ)) → Ext_{H}^{e−1}(S_{λ}, M^{⋄}).

(b) In particular,

(i) Ext_{S_{q}(r,r)}^{i}(L(μ), Δ(λ)) ≅ Ext_{H}^{i}(D_{μ′}, S_{λ′}); (i′) Ext_{S_{q}(r,r)}^{i}(∇(λ), L(μ)) ≅ Ext_{H}^{i}(D_{μ′}, S_{λ′}); (ii) Ext_{S_{q}(r,r)}^{i}(Δ(μ), Δ(λ)) ≅ Ext_{H}^{i}(S_{μ′}, S_{λ′}); (iii) Ext_{S_{q}(r,r)}^{i}(∇(μ), Δ(λ)) ≅ Ext_{H}^{i}(S_{μ}, S_{λ′}^{Φ}); (iv) Ext_{H}^{i}(S_{λ}, Y_{μ}) = 0 provided 0 < i ≤ e − 3.

Proof. Because T is an injective S_{q}(r, r)-module, there exists a spectral sequence

E_{2}^{p,q} = Ext_{S_{q}(r,r)}^{p}(M, Ext_{H}^{q}(S_{λ}, T)) ⟹ Ext_{H}^{p+q}(S_{λ}, M^{⋄}).

by Cor. 1.3. Part (a) then follows immediately from (4.4). Finally, (b) is obtained by specializing M to appropriate modules, except that (i′) is obtained from (i) by applying the duality D_{S_{q}(r,r)}. □

When q = 1 and H = k𝔖_{r}, Thm. 4.6 improves slightly upon [KN] because there is no assumption that p > 3. As pointed out above and in the introduction, this paper was in part inspired by [KN], especially the formulation in this section of the above theorem. Nevertheless, we take this opportunity to point out several minor errors in [KN]. Their version of Thm. 4.6(b) had five parts. Their part (iv) which claims that Ext_{k𝔖_{r}}^{i}(Y_{μ}, S_{λ}) = 0 is not true for i = 1. For example, Ext_{k𝔖_{5}}^{1}(Y_{(5)}, S_{(4,1)}) ≠ 0 if p = 5. (See also [HN, §4].) Their part (v) is our (iv) with the necessary additional hypothesis that i > 0. (We note that the modules that we denote by D_{μ}, Y_{μ}, and S_{μ} are denoted D^{μ}, Y^{μ}, and S^{μ} in [KN].) Finally, [KN] asserts (again when q = 1 and H = k𝔖_{r}) that the isomorphism in Thm. 4.6(a) holds for i = e − 2. This assertion is based on [KN, Prop. 2.3], which has both an incorrect statement and proof. However, the problem is only at the edge of the vanishing range, which we correct with a 5-term exact sequence, particularly in Thm. 4.6(a) and Cor. 1.3(b).

We note specifically the following immediate consequence of the above theorem.

Corollary 4.7. Assume e > 2 and let 0 ≤ i ≤ e − 3. Then

Ext_{S_{q}(r,r)}^{i}(∇(λ), L(μ)) ≅ Ext_{H}^{i}(S_{λ}, D_{μ′}^{Φ}).

Using Thm. 3.8 directly rather than Thm. 4.5 gives more precise information in cohomological degree e − 2. Thus, the result below improves upon Thm. 4.6(biii).

Proposition 4.8. Assume e > 2. For N ∈ mod–H, 0 ≤ i ≤ e − 2, and λ ∈ Λ^{+}(r),

Ext_{S_{q}(r,r)}^{i}(∇(λ), N^{⋄}) ≅ Ext_{H}^{i}(N, S_{λ′}^{Φ}).

Proof. By Thm. 3.8, there is a resolution 0 → 𝔑_{•}(S_{λ′})^{⋄} → ∇(λ) → 0. When 0 ≤ i ≤ e − 2, 𝔑_{i}(S_{λ′}) is projective, so that 𝔑_{i}(S_{λ′})^{⋄} is projective. For all i, Prop. 1.2 implies that Hom_{S_{q}(r,r)}(𝔑_{i}(S_{λ′})^{⋄}, N^{⋄}) ≅ Hom_{H}(N, 𝔑_{i}(S_{λ′})^{⋄⋄}). Because (−)^{⋄} = Hom_{S_{q}(r,r)}(−, T) is an exact functor and because ∇(λ)^{⋄} ≅ S_{λ′}^{Φ} by (2.0.16), 0 → S_{λ′}^{Φ} → 𝔑_{•}(S_{λ′})^{⋄⋄} → 0 is a resolution of S_{λ′}^{Φ} which is injective in degrees ≤ e − 2. Thus, the proposition follows in case i < e−2 (where one could also deduce it from Thm. 4.6). To see what happens when i = e − 2, consider the commutative diagram

Hom_{S_{q}(r,r)}(𝔑_{e−2}(S_{λ′})^{⋄}, N^{⋄}) \overline{}Hom_{S_{q}(r,r)}(𝔑_{e−1}(S_{λ′})^{⋄}, N^{⋄})
[figure]
Hom_{R}(N, 𝔑_{e−2}(S_{λ′})^{⋄⋄})  \overline{}Hom_{H}(N, 𝔑_{e−1}(S_{λ′})^{⋄⋄})

in which the vertical maps are isomorphisms. The kernel of the top horizonal map identifies with the kernel of the map Hom_{S_{q}(r,r)}(𝔑_{e−2}(S_{λ′})^{⋄}, N^{⋄}) → Hom_{S_{q}(r,r)}(P, N^{⋄}), if P is the projective cover of the image of the map 𝔑_{e−1}(S_{λ′})^{⋄} → 𝔑_{e−2}(S_{λ′})^{⋄}. Similarly, the kernel of the bottom horizonal map identifies with the kernel of the map

Hom_{H}(N, 𝔑_{e−2}(S_{λ′})^{⋄⋄}) → Hom_{H}(N, I),

if I is the injective envelope of the image of the map 𝔑_{e−2}(S_{λ′})^{⋄⋄} → 𝔑_{e−1}(S_{λ′})^{⋄⋄}. It now follows that Ext_{S_{q}(r,r)}^{e−2}(∇(λ), N^{⋄}) ≅ Ext_{H}^{e−2}(N, S_{λ′}^{Φ}), as required. □

We also have the following result.

Theorem 4.9. Assume that e > 2 and that 0 ≤ i ≤ e − 3. Then (a) For any M ∈ S_{q}(r, r)–mod,

Ext_{S_{q}(r,r)}^{i}(M, X(λ)) ≅ Ext_{H}^{i}(Y_{λ′}^{Φ}, M^{⋄}) = Ext_{H}^{i}(FM, Y_{λ′}^{Φ}).

Also, there is an exact sequence

0 → Ext_{S_{q}(r,r)}^{e−2}(M, X(λ)) → Ext_{H}^{e−2}(Y_{λ′}^{Φ}, M^{⋄}) → Hom_{S_{q}(r,r)}(M, Ext_{H}^{e−2}(Y_{λ′}^{Φ}, T))

→ Ext_{S_{q}(r,r)}^{e−1}(M, X(λ)) → Ext_{H}^{e−1}(Y_{λ′}^{Φ}, M^{⋄}).

(b) In particular,

(i) Ext_{S_{q}(r,r)}^{i}(L(μ), X(λ)) ≅ Ext_{H}^{i}(Y_{λ′}^{Φ}, D_{μ′}^{Φ}) ≅ Ext_{H}^{i}(Y_{λ′}, D_{μ′}); (ii) Ext_{S_{q}(r,r)}^{i}(∇(μ), X(λ)) ≅ Ext_{H}^{i}(Y_{λ′}^{Φ}, S_{μ′}^{Φ}) ≅ Ext_{H}^{i}(S_{μ}, Y_{λ′}^{Φ}); (iii) Ext_{H}^{i}(Y_{μ}, Y_{λ}) = 0 for i > 0; (iv) Ext_{H}^{i}(Y_{μ}^{Φ}, Y_{λ}) = 0 for i > 0.

Proof. We use Lemma 1.3 again with N = Y_{λ′}^{Φ}. Because Y_{λ′}^{Φ} has a Specht filtration, (4.4) implies that Ext_{H}^{i}(Y_{λ′}^{Φ}, T) = 0 for 0 < i ≤ e − 3, so (a) follows. Part (b) is obtained formally by taking the evident particular modules M. □

Thm. 3.7 can also be used to prove Thm. 4.9 directly, providing another argument for Thm. 4.5. Using the resolution (3.7.1) gives the following variation on the above result. We omit the proof, which is entirely similar to that of Prop. 4.8. Observe that for N = S_{μ} this result improves lightly on Thm. 4.9(bii).

Proposition 4.10. Assume e > 2. Then for N ∈ mod–H, λ ∈ Λ^{+}(r) and 0 ≤ i ≤ e−2, we have:

Ext_{S_{q}(r,r)}^{i}(X(λ), N^{⋄}) ≅ Ext_{H}^{i}(N, Y_{λ′}^{Φ}).

5. Cohomology, II: inequalities.

Let H = k ⊗_{Z} H̃ be the Hecke algebra of type A_{r−1} over an algebraically closed field k. We make no restriction on e > 1 (2.0.17). Often we write A for S_{q}(r, r). We will work with the Schur functor F : A–mod → H–mod; cf. Remark 1.1. Thus, if M ∈ A–mod, then FM is a left H-module, while M^{⋄} is a right H-module.

The following result does not use Thm. 3.4 and provides an inequality on dimensions of certain Ext_{H}^{•}-groups. The hypothesis is at least easy to check for i = 1, where the result essentially coincides with [DEN, Thm. (4.3)]. In fact, the theorem below (or at least its proof) extends [DEN, Thm. (4.3)] to arbitrary cohomological degrees i ≥ 0. Some remarks concerning improvements in the hypothesis are made following the proof.

Theorem 5.1. Suppose M ∈ S_{q}(r, r)–mod and for an integer j ≥ 0, consider the condition

ω ∉ Λ_{e-res}^{+}(r) ⟹ Ext_{S_{q}(r,r)}^{j}(M, L(ω)) = 0. (5.1.1)

Let P_{•} ↠ M be a minimal projective resolution. Fix an integer i. Suppose that (5.1.1) holds for j = i − 1. Then

(5.1.2) dim Ext_{S_{q}(r,r)}^{i}(M, L(ν)) ≤ dim H^{i}(Hom_{H}(FP_{•}, D_{ν′}^{Φ}))

for any ν ∈ Λ_{e-res}^{+}(r). If (5.1.1) holds, in addition, for all j < i, then

(5.1.3) dim Ext_{S_{q}(r,r)}^{i}(M, L(ν)) ≤ dim Ext_{H}^{i}(FM, D_{ν′}^{Φ})

for any ν ∈ Λ_{e-res}^{+}(r).

Proof. Consider a minimal projective resolution

⋯ → P_{1} → P_{0} → M → 0 (5.1.4)

of M. Minimality implies that the differentials in the complex Hom_{A}(P_{•}, L(ν)) are all zero. Moreover, any nonzero map g : P_{i} → L(ν) is surjective, thus determines a nonzero surjection Fg : FP_{i} ↠ FL(ν). So

dim Ext_{A}^{i}(M, L(ν)) = dim Hom_{A}(P_{i}, L(ν))

≤ dim Hom_{H}(FP_{i}, FL(ν)). (5.1.5)

Also, FL(ν) ≅ D_{ν′}^{Φ} by Lemma 2.2.

Assume that (5.1.1) holds for j = i − 1. Then we claim that Fg is not a coboundary in the complex Hom_{H}(FP_{•}, FL(ν)). Otherwise, Fg is a composite FP_{i} → FP_{i−1} → FL(ν), where the map FP_{i} → FP_{i−1} in this composite equals Fh, with h : P_{i} → P_{i−1} the differential in (5.1.4). Our hypothesis (5.1.1) guarantees that all indecomposable components of P_{i−1} have heads L(ω) with ω ∈ Λ_{e-res}^{+}(r). For such a component P (ω), the component FP (ω) ≅ Y_{ω} of FP_{i−1} is also projective, with head FL(ω) ≅ D_{ω′}^{Φ} by Lemma 2.1(b) and Lemma 2.2. That is, F(head(P_{i−1})) ≅ head(FP_{i−1}). However, the composite of h with the map P_{i−1} ↠ head(P_{i−1}) is zero, because of minimality. Thus, any composite FP_{i−1} → FL(ν) with Fh is zero. Thus, Fg is not a coboundary, as claimed.

Consider the complex Hom_{H}(FP_{•}, FL(ν)) which contains FHom_{A}(P_{•}, L(ν)) as a subcomplex with zero differentials. If (5.1.1) holds for j = i − 1, the paragraph above implies that no nonzero element of FHom_{A}(P_{i}, L(ν)) is a coboundary in the larger complex. Hence, by (5.1.5), (5.1.2) holds, as required.

Now assume that (5.1.1) holds for all j < i. Then for any such j, all the indecomposable components of P_{j} have heads L(ω) with ω ∈ Λ_{e-res}^{+}(r). Letting K = Im(P_{i} → P_{i−1}), it follows that 0 → FK → FP_{i−1} → ⋯ → FP_{0} → FM → 0 is a resolution in which each FP_{j}, j < i, is a projective H-module. A standard dimension shifting argument shows that

H^{i}(Hom_{H}(FP_{•}, FL(ν))) ≅ \frac{Hom_{H}(FK, FL(ν))}{Im(Hom_{H}(FP_{i−1}, FL(ν)) → Hom_{H}(FK, FL(ν)))}
≅ Ext_{H}^{i}(FM, FL(ν)),

as required. □

Remark 5.2. The result can be sharpened, for a fixed L(ν), using Remark 1.1 and Prop. 1.2: In (5.1.1), we replace “ω ∉ Λ_{e-res}^{+}(r)” by “ω ∉ Λ_{e-res}^{+}(r) and [L(ν)^{⋄⋄} : L(ω)] ≠ 0”. This version guarantees that Hom_{H}(FP (ω), FL(ν)) = 0 for any indecomposable component P (ω) of P_{j} with ω unrestricted: In general, dim Hom_{H}(FP (ω), FL(ν)) = dim Hom_{H}(L(ν)^{⋄}, P (ω)^{⋄}) = dim Hom_{H}(P (ω), L(ν)^{⋄⋄}) = [L(ν)^{⋄⋄} : L(ω)]. We also note that, if e > 2, the condition [L(ν)^{⋄⋄} : L(ω)] ≠ 0 implies that ω ≤ τ, where D_{ν′}^{Φ} ≅ D_{τ}. This is because the natural surjection S_{τ} ↠ L(ν)^{⋄} ≅ D_{τ} provides an inclusion L(ν)^{⋄⋄} ⊆ S_{τ}^{⋄} ≅ Δ(τ), using Lemma 2.4. Finally, if q = 1 and if L(ν)^{⋄} is completely splittable as in [KN], then [KN] shows (in our language) that L(ν)^{⋄⋄} ≅ L(ν), so that no additional hypothesis is required for the inequality (5.1.3) for any i.

As an application, we have the following result. If e > 2, then (b) is an equality by Thm. 4.6(bi′). Parts (a), (c) are also proved in [DEN, (4.3)].

Corollary 5.3. Let λ, ν ∈ Λ^{+}(ν) with ν ∈ Λ_{e-res}^{+}(r). Then:

(a) If λ ∈ Λ_{e-res}^{+}(r),

dim Ext_{S_{q}(r,r)}^{1}(Δ(λ), L(ν)) ≤ dim Ext_{H}^{1}(S_{λ′}^{Φ}, D_{ν′}^{Φ}) = dim Ext_{H}^{1}(D_{ν′}^{Φ}, S_{λ}).

(b) For λ arbitrary, dim Ext_{S_{q}(r,r)}^{1}(∇(λ), L(ν)) ≤ dim Ext_{H}^{1}(S_{λ}, D_{ν′}^{Φ}).

(c) If λ ∈ Λ_{e-res}^{+}(r),

dim Ext_{S_{q}(r,r)}^{1}(L(λ), L(ν)) ≤ dim Ext_{H}^{1}(D_{λ′}^{Φ}, D_{ν′}^{Φ}) = dim Ext_{H}^{1}(D_{λ′}, D_{ν′}).

Proof. For ω ∈ Λ^{+}(r)\Λ_{e-res}^{+}(r), Hom_{A}(Δ(λ), L(ω)) = 0 if λ ∈ Λ_{e-res}^{+}(r). Hence, the hypothesis of the theorem holds for M = Δ(λ), λ ∈ Λ_{e-res}^{+}(r), with i = 1, so (a) follows since FΔ(λ) ≅ S_{λ}^{∗} ≅ S_{λ′}^{Φ}. The proof of (c) is similar.

To see (b), first observe that Δ(λ) is a submodule of T by [CPS4, (3.2.2d)]. Since T is self-dual, there is a surjective map T ↠ ∇(λ). Hence, if L(ω) lies in the head of ∇(λ), the projective A-module P (ω) is a direct summand of T. But T = H^{⋄}, so P (ω) = Y_{ω}^{⋄} for some H-projective Young module Y_{ω}. Hence, ω ∈ Λ_{e-res}^{+}(r) by Lemma 2.1(b). Thus, the hypothesis of the theorem holds. □

If one knows the quantum James result, observed in Thm. 4.3 for e > 2, (b) can also be derived from [DEN, (4.3)]. Our argument for (b) shows that the quantum James result holds even if e = 2. Also, the i = 1 case of (5.1) can be deduced from a Grothendieck spectral sequence as in Thm. 4.6. We do not have a similar argument for i > 1.

6. Cohomology, III: smaller q-Schur algebras and specific calculations.

So far, we have only considered the cohomology of q-Schur algebra S_{q}(r, r). Now we take up the cohomology of S_{q}(n, r) for general n. For n ≤ m, there is a natural exact quotient functor j^{∗} : S_{q}(m, r)–mod → S_{q}(n, r)–mod. In particular, if n ≥ r, j^{∗} : S_{q}(n, r)–mod ⟶ S_{q}(r, r)–mod is well-known to be a category equivalence. This leaves the relationship of the categories S_{q}(n, r)–mod and S_{q}(r, r)–mod when n < r.

Fix r. Let A_{n} denote S_{q}(n, r) with A = A_{r}. Write objects associated with A_{n} with the subscript n, e. g., L_{n}(λ), Δ_{n}(λ) and ∇_{n}(λ) are the irreducible, standard and costandard modules for A_{n}. The main result is the following:

Theorem 6.1. Let n ≤ r and let j^{∗} : A–mod → A_{n}–mod be the natural exact quotient functor. For any finite dimensional A = S_{q}(r, r)-module M and λ ∈ Λ^{+}(n, r),

Ext_{A}^{•}(Δ(λ), M) ≅ Ext_{A_{n}}^{•}(Δ_{n}(λ), j^{∗}M) and Ext_{A}^{•}(M, ∇(λ)) ≅ Ext_{A}^{•}(j^{∗}M, ∇_{n}(λ)).

(Recall from [CPS6, §2] that j^{∗}L(λ) = L_{n}(λ), j^{∗}Δ(λ) = Δ_{n}(λ) and j^{∗}∇(λ) = ∇_{n}(λ) for λ ∈ Λ^{+}(n, r). If λ ∉ Λ^{+}(n, r), then j^{∗}L(λ) ≅ j^{∗}Δ(λ) ≅ j^{∗}∇(λ) = 0.)

Proof. We work with the derived category D^{b}(A–mod) as in [CPS6, §2]. Let j^{∗} : D^{b}(A–mod) → D^{b}(A_{n}–mod) be the derived functor of j^{∗}. Then j^{∗} admits a left adjoint j_{∗} satisfying j_{!}Δ_{n}(λ) ≅ Δ(λ) for all λ ∈ Λ^{+}(n, r) by [CPS6, Lemma 3]. Thus,

Ext_{A}^{•}(Δ(λ), M) ≅ Hom_{D^{b}(A–mod)}^{•}(Δ(λ), M) ≅ Hom_{D^{b}(A–mod)}^{•}(j_{!}Δ_{n}(λ), M)
≅ Hom_{D^{b}(A_{n}–mod)}^{•}(Δ_{n}(λ), j^{∗}M) ≅ Ext_{A_{n}}^{•}(Δ_{n}(λ), j^{∗}M).

The second assertion holds by duality. □

The result allows us to transform cohomological information from the smaller and more tractable (especially if n ≤ e) category S_{q}(n, r)–mod, n < r, to the category S_{q}(r, r)–mod. Many relationships of S_{q}(n, r)-cohomology with H-cohomology do not apply directly in the n < r case, but do apply to the transferred information.

For simplicity, consider the q = t̄ = 1 case, i. e., H ≅ k𝔖_{r}. Of course, we assume that the characteristic p of the algebraically closed field k is positive. Our applications will critically involve symmetric groups 𝔖_{r} and Schur algebras S(n, r) with n ≤ p, an hypothesis of the Lusztig conjecture.

Another important ingredient here will be Thm. 6.3, extending to cohomology a decomposition number method of Erdmann [E]. This requires some notation and a lemma. Let F : GL_{n}(k) → GL_{n}(k), g = [a_{i,j}] ↦ g^{F} = [a_{i,j}^{p}], be the Frobenius morphism. Set G_{1} = Ker(F). Given a rational GL_{n}(k)-module V, V^{(1)} denotes the rational GL_{n}(k)-module obtained from V by making GL_{n}(k) act through F. For any positive integer r, identify S(n, r)–mod with the full subcategory of polynomial GL_{n}(k)-modules which are homogeneous of degree r. For V, W ∈ S(n, r)–mod,

Ext_{S(n,r)}^{•}(V, W) ≅ Ext_{GL_{n}(k)}^{•}(V, W). (6.1.1)

See [D1] or [CPS1].

Let ρ_{r} = (r−1, r−2, ⋯, 1, 0) ∈ Λ^{+}(\frac{r(r−1)}{2}). Let St = L((p−1)ρ)_{r} ∈ S(r, (p−1)\frac{r(r−1)}{2}) be the Steinberg module. As a G_{1}-module, St is self-dual projective. For λ ∈ Λ^{+}(s_{1}) and μ ∈ Λ^{+}(s_{2}), define λ + μ ∈ Λ^{+}(s_{1} + s_{2}) by setting (λ + μ)_{i} = λ_{i} + μ_{i}, ∀i.

Inspired by Erdmann [E], define d : Λ^{+}(r) → Λ^{+}(pr + (p − 1)\frac{r(r−1)}{2}) by

d(λ) = (pλ′ + (p − 1)ρ_{r})′ ∈ Λ^{+}(pr + (p − 1)\frac{r(r − 1)}{2}), (6.1.2)

λ ∈ Λ^{+}(r). As noted in [E], d(λ)′ ∈ Λ_{p–reg}^{+}(s), so d(λ) is always p-restricted, recorded in (c) below.

Lemma 6.2. Let λ ∈ Λ^{+}(r) (r arbitrary) and set s = pr + (p − 1)\frac{r(r−1)}{2}. Then: (a) Regarding Δ(λ) as a GL_{r}(k)-module, Δ(λ)^{(1)} ⊗ St ≅ Δ(pλ + (p − 1)ρ_{r}) ∈ S(r, s).

(b) Regarding the tilting module X(λ) as a GL_{r}(k)-module, X(λ)^{(1)} ⊗ St ≅ X(pλ + (p − 1)ρ_{r}) ∈ S(r, s)–mod.

(c) d(λ) ∈ Λ_{p–res}^{+}(s).

Proof. For (a), see [A1, (2.5)], and for (b), see [E, (2.1)]. □

Theorem 6.3. For any μ ∈ Λ^{+}(r), we have for all i ≥ 0,

dim Ext_{S(r,r)}^{i}(Δ(λ), L(μ)) = dim Ext_{S(s,s)}^{i}(Δ(d(λ)), L(d(μ))).

(Using (6.1.1), these groups can be reinterpreted in terms of GL_{r}(k)- and GL_{s}(k)- cohomology, respectively.)

Proof. Form the complete tilting module X = ⨁_{λ∈Λ^{+}(r)} X(λ) for S(r, r). If E = End_{S(r,r)}(X), E–mod is a HWC with poset (Λ^{+}(r), ⊴^{op}) [R]. There is an equivalence of categories G : E–mod →^{∼} S(r, r)–mod in which GΔ_{E}(λ) = Δ(λ′), ∀λ ∈ Λ^{+}(r). See [D2] or [DPS2, §7]. (The equivalence here is due to Donkin. It implies that GL_{E}(λ) ≅ L(λ′), GX_{E}(λ) ≅ X(λ′), etc.) Let

P_{•} → Δ(λ) → 0 (6.3.1)

be a minimal projective resolution in S(r, r)–mod. Consider the functor

𝔱 = Hom_{S(r,r)}(−, X) : S(r, r)–mod → E–mod.

Since the kernels in the resolution (6.3.1) have Δ-filtrations by Lemma 1.4(b), 0 → 𝔱Δ(λ) → 𝔱P_{•} is still a resolution, using Lemma 1.4(a) this time. In addition, each 𝔱P_{i} is a tilting module for E–mod. Making using of the equivalence of categories G above, we obtain a resolution 0 → Δ(λ′) → X_{•} in which each X_{i} is a tilting module for S(r, r). Now apply the exact functor S(r, r)–mod → S(r, s)–mod, M ↦ M^{(1)} ⊗ St, to obtain a resolution 0 → Δ(d(λ)′) → X′_{•}, where each X′_{i} is a direct sum of indecomposable tilting modules X(μ) for μ ∈ Λ^{+}(r, s) by Lemma 6.2. In addition, the cokernels of this resolution have Δ-filtrations by Lemma 1.4(b) (working from the right to the left in the complex). Letting X′ be a complete tilting module for S(r, s)–mod and applying the functor Hom_{S(r,s)}(−, X′) yields a resolution of Hom_{S(r,s)}(Δ(d(λ)′), X′) in E′-mod, where E′ = End_{S(r,s)}(X′). Using [D2] or [DPS2, (7.7)], E′–mod identifies with the full subcategory S(s, s)–mod[Λ^{+}(r, s)′] of S(s, s)–mod. The functor Hom_{S(r,s)}(−, X′) is compatible with the analogous functor for S(s, s)–mod, taking the quotient category S(r, s)–mod to this full subcategory, and providing a projective resolution

P′_{•} → Hom_{S(r,s)}(Δ(d(λ)), X′) ≅ Δ(d(λ)) → 0. (6.3.2)

Furthermore, because (6.3.1) is minimal, (6.3.2) is also minimal: Essentially, the process which led to (6.3.2) is entirely reversible. If (6.3.2) is not minimal, it has a nonzero acyclic projective complex as a summand, with some subset of the same associated weights for the constituent indecomposable factors as for (6.3.2). This acyclic complex of projective modules is obviously bounded and becomes a bounded acyclic complex of tilting modules in the previous step–the inverse functor is also expressed by a Hom to a complete tilting module. The process of applying a Frobenius twist and tensoring with a Steinberg module is inverted by applying the functor Hom_{G_{1}}(St, −), and then factoring out the action of G_{1}, again giving a bounded acyclic complex of tilting modules as a nonzero summand. Finally, this gives a nonzero acyclic complex of projectives as a summand of (6.3.1), a contradiction. Thus, (6.3.2) must be minimal. For a given i, dim Ext_{S(r,r)}^{i}(Δ(λ), L(μ)) equals the number of times that the projective indecomposable module P (μ) occurs as a direct summand of P_{i} in (6.3.1), which, by construction, equals the number of times that P (d(λ)) ∈ S(s, s)–mod[Λ^{+}(r, s)′] occurs as a direct summand of P′_{i}. In turn this number equals dim Ext_{S(s,s)}^{i}(Δ(d(λ)), L(d(μ)). (In the context of [CPS6, §2], use the fact that i_{∗} is a full embedding of derived categories.) □

Remarks 6.4. (a) Although not needed here, observe that there is an equality [Δ(λ) : L(μ)] = [Δ(d(λ)) : L(d(μ))] of decomposition numbers. The proof, along the lines of [E], is similar to that of Thm. 6.3: equate [Δ(λ) : L(μ)], via Brauer-Humphreys reciprocity and duality, to the multiplicity of Δ(μ) in P (λ). After tilting via the functor Hom_{S(r,r)}(−, X), this equals the multiplicity of Δ(λ′) in the tilting module X(μ′). That multiplicity is preserved under the functor M ↦ M^{(1)} ⊗ St and then under another tilting functor. Finally, apply Brauer-Humphreys reciprocity again.

Another proof of this result can be obtained using Thm. 6.3 and an Euler characteristic argument (as given, for example, in [CPS6, §3]), plus with the observation that the tilting functor Hom_{S(r,r)}(−, X) preserves blocks.^{5}

(b) [CPS6, §5] proves that Thm. 6.3 follows from an appropriate Morita equivalence (which also implies the above equality of decomposition numbers).

Put G = SL_{n}(k), and let T be the maximal torus consisting of diagonal matrices of determinant 1. Then T ⊂ T′, the maximal torus of GL_{n}(k) consisting of all nonsingular diagonal matrices. Let X(T) (resp., X(T′)) be the character group on T (resp., T′). Let X^{+}(T) be the set of dominant weights on T with respect to the Borel subgroup B of upper triangular unimodular matrices. Identify X(T) (resp., X^{+}(T)) with sequences a = (a_{1}, ⋯, a_{n−1}) of integers (resp., non-negative integers): if s = diag(s_{1}, ⋯, s_{n}) ∈ T, a(s) = s_{1}^{a_{1}+···+a_{n−1}}s_{2}^{a_{2}+···+a_{n−1}} ⋯ s_{n−1}^{a_{n−1}}. For 1 ≤ i ≤ n − 1, the ith fundamental dominant weight a_{i} is defined by a_{i}(s) = s_{1} ⋯ s_{i}. Also, X(T′) identifies with sequences of λ = (λ_{1}, ⋯, λ_{n}) of integers, and restriction res : X(T′) → X(T) satisfies res(λ) = (a_{1}, ⋯, a_{n−1}), with a_{i} = λ_{i} − λ_{i−1}, 1 ≤ i ≤ n − 1, putting λ_{0} = 0. Thus, res(λ) ∈ X^{+}(T) if λ is a partition. We denote res(ρ_{n}) simply by ρ; as a weight on T, ρ equals one-half the sum of the positive roots. Then a ∈ X^{+}(T) is regular provided that (a+ρ, α^{∨}) ≢ 0 mod p for all positive roots α. Here α^{∨} denotes the coroot associated to α and (−, −) is the natural pairing between characters and coroots.

5Let |μ| = r and |d(μ)| = s. The s-homogeneous blocks of polynomial GL_{s}(k)-modules containing the irreducible module L(d(μ)) contains only irreducible modules L(d(ω)), where ω ∈ Λ^{+}(r). In particular, all such weights d(ω) are p-restricted!

Given a ∈ X^{+}(T), let Δ(a), ∇(a), L(a) be the standard, costandard, and irreducible G-module of highest weight a. If λ is a partition such that res(λ) = a, then Δ(λ)|_{G} ≅ Δ(a), ∇(λ)|_{G} ≅ ∇(a), and L(λ)|_{G} ≅ L(a). Here Δ(λ), etc. are modules for GL_{n}(k); they are lie in the category S(n, r)–mod if r = |λ|.

Now assume that that p ≥ n (= Coxeter number h of G = SL_{n}(k)). The Jantzen region Γ ⊆ X(T) consists of all weights a ∈ X(T) satisfying (a + ρ, α_{0}^{∨}) ≤ p(p − h + 2), where α_{0} = (1, 0, ⋯, 0, 1) ∈ X(T) is the maximal (short) root. (The notion makes sense for any semisimple group.) Assume that Lusztig’s conjecture for G holds; for a regular dominant weight a in the Janzten region, this conjecture provides an explicit character formula for the irreducible G-module L(a). For the precise statement, valid for all semisimple groups, see [L] (or [A2, (1.4)]).

For regular dominant weights a, b ∈ Γ, the groups Ext_{G}^{•}(Δ(a), L(b)) are determined by Kazhdan-Lusztig polynomials P_{x,y} ∈ ℤ[v, v^{−1}] associated to the affine Weyl group W_{p} attached to G; cf. [A2, (2.1.2)].^{6} To describe this calculation in the essential case, assume that a, b both belong to the orbit O of −2ρ under W_{p}. For a ∈ O ∩ X^{+}(T), let w_{a} ∈ W_{p} satisfy w_{a} · (−2ρ) = a. Then we have, with these notations and assumptions,

(6.4.1)  ∑_{i=0}^{∞} dim Ext_{G}^{i}(Δ(a), L(b))v^{i} = v^{ℓ(w_{b})−ℓ(w_{a})}P̄_{w_{a},w_{b}},

where P̄_{w_{a},w_{b}} is obtained from P_{w_{a},w_{b}} by replacing v by v^{−1}.

Example 6.5. As discussed in [S], the Lusztig conjecture for the characters of the irreducible G = SL_{5}(k)-modules holds when p = 5 or 7. Let p = 7, a = 0 and b = 2a_{1} + 5a_{2} + 5a_{3} + 2a_{4}. Then P_{w_{a},w_{b}} = 1 + v^{0} + 4v^{2} + 5v^{4} + 4v^{6} + 2v^{8} and ℓ(w_{b}) − ℓ(w_{a}) = 9. See [S] and the tables in www.math.virginia.edu/ ∼lls2l. Thus, dim Ext_{SL_{5}(k)}^{1}(Δ(a), L(b)) = 2. At the level of GL_{5}(k), consider the partitions μ = (7, 7, 7, 7, 7) and λ = (14, 12, 7, 2, 0) of 35. Then res(μ) = a and res(ξ) = b. Thus, dim Ext_{GL_{5}(k)}^{1}(Δ(μ), L(λ)) = dim Ext_{S(5,35)}^{1}(Δ(μ), L(λ)) = 2.

6Here v = \sqrt{t} in the notation of this paper. Kazhdan-Lusztig polynomials are usually written in terms of “q”, which is our t = v^{2}.

However, μ is not 7-restricted, so Cor. 5.3(a) does not apply. But Thm. 6.3 and Thm. 6.1 imply that dim Ext_{S(305,305)}^{1}(Δ(d(μ)), L(d(λ)) = 2. The point is that the Lusztig conjecture does not apply to SL_{305}(k) since p = 7 is less than the Coxeter number 305 of SL_{305}(k). The same difficulty would have occurred if μ were restricted, and we were able to work with S(35, 35) without using d(μ).) Now both d(λ) and d(μ) are 7-restricted, so using Cor. 5.3(a), we conclude that

dim Ext_{k𝔖_{305}}^{1}(S_{d(μ)′}, D_{d(λ)′}) ≥ 2.

The introduction to [DEN] makes a reference to a conjecture of Kleshchev and Guralnick that dim Ext_{k𝔖_{r}}^{1}(S_{(r)}, D_{ν′}) ≤ 1, for any ν ∈ Λ_{e-res}^{+}(r). Example 6.5 shows that (r) may not be replaced by an arbitrary p-regular partition in this conjecture.

Remarks 6.6. (a) When k = ℂ and q is a primitive ℓth root of unity, the approach of Thm. 6.3 is not available. However, the rest of our discussion still applies, and the function d(μ) is not required if μ is e = ℓ-restricted. Moreover, the analogue of Lusztig’s conjecture holds for the quantum enveloping algebra U_{q}; hence, for regular λ, μ (in the sense of alcove geometry) the groups Ext_{S_{q}(n,r)}^{•}(Δ(λ), L(μ)) are determined by means of the coefficients of Kazhdan-Lusztig polynomials. Thus, the method described above can be applied in this case to obtain lower bounds on certain Ext_{H}^{1}-groups. In fact, a lower bound on Ext_{S_{q}(n,r)}^{1}(Δ(λ), L(μ)), μ > λ, and thus a corresponding lower bound on Ext_{H}^{1}(S_{μ′}, D_{λ′}) can be obtained even when λ, μ are not regular. The idea is to interpret Ext_{S_{q}(n,r)}^{1}(L(μ), L(λ)) in terms of the structure of Δ(μ), and use Jantzen translation arguments. This procedure may be applied for all e ≥ 1 by passing to the affine Lie algebra [KL], [Ku]. Another relevant reference is [CPS3, Cor. 3.6(b) & Thm. 4.3].

(b) There is some substitute for Erdmann’s decomposition number method, which is due to Leclerc [Le, Thm. 1]. However, this result seems hard to apply directly to cohomology, though [Le, Thm. 2], an equality on certain polynomials, does likely have cohomological content. It might be illuminated by passing to the affine Lie algebra [KL] and proving a graded version of Leclerc’s theorem on decomposition numbers.

7. Appendix: the Lusztig conjecture and the symmetric group.

Let G be a semisimple, simply connected group over an algebraically closed field k of characteristic p ≥ h. (h = the Coxeter number of G.) Let O′ be the set of dominant weights which lie in the W_{p}-orbit of −2ρ and in the Janzten region Γ as defined above.

The Lusztig conjecture for the characters of the irreducible G-modules with highest weight in O′ is equivalent [CPS3, (5.4b)] to the assertion that, given a, b ∈ O′,

(7.0.1) ℓ(w_{a}) − ℓ(w_{b}) ≡ 0 (mod 2) ⟹ Ext_{G}^{1}(Δ(a), L(b)) = 0,

in the notation of the previous section. Also, we can assume that a < b in the sense that b − a is a sum of positive roots.

Let G = SL_{n}(k). There exists a positive integer r with the following property: for each a ∈ O′, there exists a partition λ_{a} ∈ Λ^{+}(n, r) such that res(λ_{a}) = a. In this case, given a, b ∈ O′, we have by Thm. 6.1,

Ext_{SL_{n}(k)}^{•}(Δ(a), L(b)) ≅ Ext_{S(n,r)}^{•}(Δ_{n}(λ_{a}), L_{n}(λ_{b})) ≅ Ext_{S(r,r)}^{•}(Δ(λ_{a}), L(λ_{b})).

Then Cor. 5.3(a) and Thm. 6.3 imply that

(7.0.2) Ext_{k𝔖_{s}}^{1}(S_{d(λ_{a})′}, D_{d(λ_{b})′}) = 0 ⟹ Ext_{SL_{n}(k)}^{1}(Δ(a), L(b)) = 0.

(Recall that s = pr + (p − 1)\frac{r(r−1)}{2}.) Thus, we have the following cohomological result:

Proposition 7.1. Let O′ = W_{p} · (−2ρ) ∩ Γ. The Lusztig conjecture holds for SL_{n}(k) provided that, for all a < b ∈ O′ such that ℓ(w_{a}) ≡ ℓ(w_{b}) mod 2, we have

Ext_{k𝔖_{s}}^{1}(S_{d(λ_{a})′}, D_{d(λ_{b})′}) = 0.

Remarks 7.2. (a) We do not know if the Ext^{1}-vanishing condition above for the symmetric groups is equivalent to the Lusztig conjecture for SL_{n}(k).

(b) As noted in Remark 6.4(a), Erdmann [E] used tilting module theory to compute the decomposition numbers for Schur algebras in terms of decomposition numbers of symmetric groups. Her work, thus, already provides one approach to attacking any SL_{n}(k)–conjecture on decomposition numbers via the representation theory of symmetric groups. The above Ext^{1}-vanishing condition, based on Thm. 6.3, just suggests another approach—certainly a related one—in the more specific case of the Lusztig conjecture, using cohomological features especially appropriate to that conjecture. In the spirit of this paper, Thm. 6.3 provides the required extension of Erdmann’s decomposition number method to cohomology.

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Department of Mathematics, University of Virginia, Charlottesville, VA 22903

Department of Mathematics, University of Virginia, Charlottesville, VA 22903