EXTENSIONS, LEVI SUBGROUPS AND CHARACTER FORMULAS

BRIAN J. PARSHALL AND LEONARD L. SCOTT

We dedicate this paper to the memory of Walter Feit

Contents

Introduction 1 Part I: A category equivalence and applications 3 1. The equivalence 3 2. Some elementary applications 6 Part II: Character formulas 9 3. The Lusztig character formula 9 4. Equivalent conditions 12 5. Type A 16 Part III: Levi subgroups and irreducible modules 21 6. A theorem of Hemmer and a generalization 21 7. An example 22 8. An Ext^{1}-isomorphism result and higher Ext^{•}-groups 23 References 27

Introduction

Let H be a Levi subgroup of a reductive algebraic group G defined over an algebraically closed field k of positive characteristic p. This paper has its origins in the authors’ attempts to understand the relationship between the cohomology of H and that of G. Usually, we concentrate on comparing Ext_{G}^{•}(M, N) to Ext_{H}^{•}(M_{H}, N_{H}), where M, N are rational G-modules and M_{H}, N_{H} are rational H-modules obtained from M, N by a natural “truncation” process. Especially important examples include the cases when M, N are taken to be standard (or Weyl) modules, costandard modules, or irreducible modules.

Contributions to the problem have been made by E. Cline [C] and S. Donkin [D1]. Their work was the starting point for our discovery in [CPS6] (with E. Cline) of an explicit category equivalence between some highest weight categories associated to the categories of rational modules for G and H. Because the methods were generally formal, they also applied to module categories for quantum enveloping algebras and, in type A, for q-Schur algebras at a root of unity. Furthermore, these results provided natural Ext-transfer results in going from certain H-modules to G-modules.

1991 Mathematics Subject Classification. Primary 17B55, 20G; Secondary 17B50.

Research supported in part by the National Science Foundation.

Part I of the present paper reviews the category equivalence described above, and gives a number of examples not explicitly considered in [CPS6]. For example, when H decomposes into a direct product of reductive subgroups, the groups Ext_{H}^{•}(M_{H}, N_{H}) can be further analyized by means of an evident application of the Künneth theorem. In type A, these results can be stated in an elegant way, using the elementary combinatorics of partitions. In this case, our formulation was inspired by the corresponding degree 0 decomposition given by Lyle-Mathas in a preprint [LM]. (More recently, A. Mathas has made a revised verison of [LM] available to us, in which they also obtain, by different methods, the Künneth decomposition for q-Schur algebras.) It should be remarked, however, that the methods of [CPS6] apply for reductive groups or quantum groups in all types. As an illustration, we very briefly consider the case of type C, which behaves like the type A case.

The results of Part II improve upon [PS, §7] which formulated a vanishing condition for symmetric group cohomology which, if true, would be sufficient to prove the Lusztig character conjecture for G = SL_{n}(k). In §4, we allow G to be of any type with p ≥ h, the Coxeter number of G. Roughly, we consider finite ideals Γ of dominant weights having the following property (†): if ϖ ∈ Γ is regular, then the character of the irreducible G-module L(ϖ) of highest weight ϖ is given by the Lusztig character formula; see (3.0.2). Lusztig’s conjecture [L] asserts that the Jantzen region Γ_{Jan} (defined in (3.0.3)) is such an ideal; however, independently of that conjecture, finite ideals Γ satisfying property (†) are easily shown to exist. In that case, various homological methods can be brought to bear on the category C_{G}[Γ] of finite dimensional rational G-modules which have composition factors L(γ), γ ∈ Γ. For example, [CPS2] establishes that Γ satisfies (†) if and only if Ext_{G}^{1}(L(λ), L(μ)) ≠ 0 whenever λ, μ ∈ Γ are regular weights which are mirror images of each other in adjacent alcoves. This result is recalled in Thm. 4.1, which also presents a useful (and new) method allowing the possible enlargement of (certain ideals) Γ by the addition of an alcove adjacent to Γ. Thm. 4.2 recasts the Ext^{1}-criterion above in terms of a condition on mappings between Ext^{1}-groups. The modules involved are no longer irreducible; indeed, all are standard modules or modules with a standard filtration.

In Section 5, Thm. 5.2 translates, in the case G = SL_{n}(k), the results of §4 into equivalent results stated entirely in symmetric group terms. In particular, we obtain a necessary and sufficient condition for the Lusztig character conjecture to hold for SL_{n}(k), stated entirely in terms of the representation theory of the symmetric group. This result is in contrast to [PS, §7], which only obtained a sufficient condition. Section 5 concludes with a discussion of the relation of Thm. 5.2 to the sufficient condition established in [PS, §7]. See Remark 5.5(b) for a brief comparison between features of our Lusztig conjecture reduction to those of Erdmann’s general computation [E2] of Schur algebra decomposition numbers in terms of those of the symmetric group.

The results in Part I for q-Schur algebras apply to Ext-groups between standard modules, or between costandard modules, or between an irreducible and a standard or costandard module. Part III takes up the question of comparing Ext-groups between irreducible G-modules with those Ext-groups between similarly parametrized irreducible H-modules. One result in this direction, for Schur algebras, has already been obtained by Hemmer [H], who proves the injectivity of the natural map ϕ_{H}^{G}(ϖ, ϖ′) : Ext_{G}^{1}(L(ϖ), L(ϖ′)) → Ext_{H}^{1}(L_{H}(ϖ), L_{H}(ϖ′)) when the dominant weights ϖ, ϖ′ lie in the same coset relative to the weight lattice of H. His result is stated in the type A formalism of row removal in Young diagrams and Schur algebras. However, in §6, Thm. 6.1 proves that Hemmer’s result holds for all types. (A similar result would hold, with the same proof, for quantum enveloping algebras.) Hemmer has asked whether the map ϕ_{H}^{G}(ϖ, ϖ′) is an isomorphism. In §7, we provide several (related) examples showing that the answer to Hemmer’s question is sometimes negative. The examples involve weights at which the Lusztig character formula fail, and thus suggest a connection between Hemmer’s question and the validity of the Lusztig character formula.

Now assume, for general G, that the Lusztig character formula holds for all irreducible modules L(ϖ) with regular highest weights ϖ in a given finite ideal Γ in the poset of dominant weights. Then, as perhaps suggested by the example in §7, Thm. 8.3 shows that ϕ_{H}^{G}(ϖ, ϖ′) is an isomorphism if ϖ, ϖ′ ∈ Γ are regular and lie in the same coset relative to the root lattice of H. Thus, Hemmer’s question does have a positive answer in this case. Moreover, we prove that the restriction map Ext_{G}^{n}(L(ϖ), L(ϖ′)) → Ext_{H}^{n}(L(ϖ), L_{H}(ϖ′)) is surjective in all degrees n. A key ingredient in these results is the study in [CPS3] of the structure of the algebra A^{!} = Ext_{A}^{•}(L_{0}, L_{0}), when A is a quasi-hereditary algebra such that A–mod has a Kazhdan-Lusztig theory; here L_{0} is the direct sum of the distinct irreducible A-modules.

Actually, Part III of this paper developed before Part II, following quite naturally upon the investigation in Part I. The relationship with the Lusztig character formula which emerged inspired the revisiting of the latter topic in Part II, though only the results up through Thm. 4.1 are really required in Part III. One interesting by-product of this investigation is a final observation, given in Cor. 8.7, showing that a form of the Lusztig conjecture holds for any Levi subgroup H, once it holds for G.

PART I: A category equivalence and applications

We first review an essential category equivalence involving Levi subgroups of reductive groups. The method has applicability to other situations, e. g., quantum groups, q-Schur algebras, as we illustrate. In addition, the equivalence will play an important role later in the paper.

1. The equivalence

Fix a reductive group G over a fixed algebraically closed field k of positive characteristic p. We will assume that the derived group of G is simply connected. Let B be a Borel subgroup of G and let T be a maximal torus of G contained in B. Let X = X(T) be the character group of T, and set X^{∨} = Hom(𝔾_{m}, T), the cocharacter group of T. There is a natural perfect pairing ⟨ , ⟩ : X × X^{∨} → ℤ ≅ End(𝔾_{α}). We let Φ ⊂ X be the root system of T in G, and Φ^{+} (resp., Π) be the set of positive (resp., simple) roots determined by B; thus, if α ∈ Φ^{+}, the corresponding one-parameter root subgroup U_{α} is a subgroup of B. For α ∈ Φ, let α^{∨} ∈ X^{∨} be the associated coroot. Now let P ⊇ B be a parabolic subgroup, having Levi factor H ⊇ T. We also refer to H as a Levi subgroup. The root system of T in H is denoted Φ_{H}, while Φ_{H}^{+} := Φ_{H} ∩ Φ^{+} is a set of positive roots and Π_{H} = Π ∩ Φ_{H} is the corresponding set of simple roots.

We let X^{+} ⊂ X (resp., X^{+(H)} ⊂ X) be the set of dominant (resp., H-dominant) weights on T. Thus, λ ∈ X^{+} (resp., λ ∈ X^{(H)+}) provided that ⟨λ, α^{∨}⟩ ∈ ℤ^{+} for all α ∈ Π (resp., α ∈ Π_{H}). Define poset structures ≤ and ≤_{H} on X by putting λ ≤ μ (resp., λ ≤_{H} μ) provided that μ−λ ∈ ℤ^{+}Φ^{+} (resp., μ−λ ∈ ℤ^{+}Φ_{H}^{+}). By [CPS6, Lemma 6], X^{+} is an ideal in the poset (X^{(H)+}, ≤_{H}), i. e., if μ ≤_{H} λ and μ ∈ X^{(H)+}, λ ∈ X^{+}, then μ ∈ X^{+}.

Let C_{G} (resp., C_{H}) be the category of finite dimensional rational G-modules (resp., H-modules). The irreducible objects in C (resp., C_{H}) are indexed by the set X^{+} (resp., X^{(H)+}). For λ ∈ X^{+}, let L(λ), Δ(λ), and ∇(λ) be the (rational) irreducible, standard, and costandard G-modules, respectively, of highest weight λ. Similarly, for σ ∈ X^{(H)+}, L_{H}(σ), Δ_{H}(σ), ∇_{H}(σ) be the analogous objects in the category of rational H-modules.

Let Γ^{+} be a finite ideal in the poset (X^{+}, ≤), and let C_{G}[Γ^{+}] be the full subcategory of C consisting of all objects in C which have composition factors L(γ), γ ∈ Γ^{+}. Thus, C_{G}[Γ^{+}] is a highest weight category with poset (Γ^{+}, ≤).^{1} If Ξ^{+} ⊆ Γ^{+} is a coideal (i. e., Γ^{+}\Ξ^{+} is an ideal), we put C_{G}(Ξ^{+}) equal to the quotient category of C_{G}[Γ^{+}] by the Serre subcategory C_{G}[Γ^{+}\Ξ^{+}]. Then C_{G}(Ξ^{+}) is a highest weight category with poset (Ξ^{+}, ≤).^{2} Let j^{∗} : C_{G}[Γ^{+}] → C_{G}(Ξ^{+}) be the natural quotient functor. For ξ ∈ Ξ^{+}, the functor j^{∗} takes the irreducible (resp., standard, costandard) object L(ξ) (resp., Δ(ξ),∇(ξ)) in C_{G}[Γ^{+}] indexed by ξ to the irreducible (resp., standard, costandard) object L_{Ξ^{+}}(ξ) (resp., Δ_{Ξ^{+}}(ξ), ∇_{Ξ^{+}}(ξ)) in C_{G}(Ξ^{+}) indexed by ξ. If γ ∈ Γ^{+}\Ξ^{+}, j^{∗} annihilates L(γ), Δ(γ), ∇(γ). These facts are all well-known general properties of highest weight categories—see, e. g., [CPS6, §2] and the references given there.

1If Γ ⊂ X, it is convenient to write C_{G}[Γ] for the full subcategory of C_{G} having objects which have composition factors L(γ), γ ∈ Γ. Thus, if Γ^{+} := Γ ∩ X^{+}, then C_{G}[Γ] = C_{G}[Γ^{+}]. Also, if Γ is an ideal in (X, ≤), Γ^{+} is an ideal in (X^{+}, ≤).

The set X^{+} in this paragraph may be replaced with the set of highest weights of irreducible modules in any block or union of blocks for the category of rational G-modules. The same discussion applies.

We will return to this topic later in §8. If will be useful to observe that if Γ is a union of blocks in an ideal, then C_{G}[Γ] is a highest weight category with poset (Γ, ≤).

2The definition of C_{G}(Ξ^{+}) does not depend on the choice of the finite ideal Γ^{+}: Suppose that Ξ^{+} is any finite subset of X^{+} which is interval closed, in the sense that, given, ξ_{1} < ξ_{2} in Ξ^{+}, any ξ ∈ X^{+} satisfying ξ_{1} ≤ ξ ≤ ξ_{2} automatically belongs to Ξ^{+}. Then Ξ^{+} is a coideal in the finite ideal Γ′ consisting of all γ ∈ X^{+} satisfying γ ≤ ω for some ξ ∈ Ξ. Thus, C_{G}(Ξ^{+}) can be taken to be the highest weight category C_{G}[Γ′]/C_{G}[Γ′\Ξ^{+}]. If Ξ^{+} is a coideal in another finite ideal Γ^{+}, it is readily verified that C_{G}(Ξ^{+}) is equivalent to C_{F} [Γ^{+}]/C_{G}[Γ^{+}\Ξ^{+}].

Fix ω ∈ X and let Ω = ω + ℤΦ_{H} be the corresponding coset of ℤΦ_{H} in X. Set Ω^{+} = Ω ∩ X^{+} and Ω^{(H)+} = Ω ∩ X^{(H)+}. Let F be a finite, nonempty subset of Ω^{+}.^{3} Let Γ_{F}^{+} denote the (finite) ideal in (X^{+}, ≤) generated by F, and put Ω_{F}^{+} = Γ_{F}^{+} ∩ Ω. It is shown in [CPS6, p. 222] that Ω_{F}^{+} is a coideal in (Γ_{F}^{+}, ≤), but an ideal in (X^{(H)+}, ≤_{H}). We consider the truncation functor

π_{Ω} : C_{G}[Γ_{F}^{+}] → C_{H}[Ω] = C_{H}[Ω^{+(H)}], M ↦ π_{Ω}M = ⨁_{τ∈Ω} M_{τ}. (1.0.1)

Here M_{τ}, τ ∈ Ω, denotes the τ-weight space of the rational G-module M. By [D1] and [CPS6, Prop. 7], π_{Ω} maps L(ω), Δ(ω), ∇(ω), ω ∈ Ω_{F}^{+} to the corresponding objects L_{H}(ω), Δ_{H}(ω), ∇_{H}(ω) in C_{H}[Ω_{F}^{+}]. (If γ ∈ Γ_{F}^{+}\Ω_{F}^{+}, π_{Ω} maps these objects to 0.) We have:

Theorem 1.1. [CPS6, Thm. 8] The functor π_{Ω} factors through the quotient morphism j^{∗} : C_{G}[Γ_{F}^{+}] → C_{G}(Ω_{F}^{+}) to produce an equivalence

C_{G}(Ω_{F}^{+}) ⟶_{∼}^{\overline{π_{Ω}}} C_{H}[Ω_{F}^{+}] (1.1.1)

of (highest weight) categories.

This means, given ω ∈ Ω_{F}^{+} and M ∈ C_{G}[Λ_{F}^{+}], there are isomorphisms

Ext_{H}^{•}(Δ_{H}(ω), π_{Ω}M) ≅ Ext_{C_{H}[Ω_{F}^{+}]}^{•}(Δ_{H}(ω), \overline{π_{Ω}}j^{∗}M)
≅ Ext_{C_{G}(Ω_{F}^{+})}^{•}(j^{∗}Δ(ω), j^{∗}M)
≅ Ext_{C_{G}[Γ_{F}^{+}]}^{•}(Δ(ω), M)
≅ Ext_{G}^{•}(Δ(ω), M).

(1.1.2)

The last isomorphism follows since, at the derived category level, C_{G}[Γ_{F}^{+}] fully embeds into the category of rational G-modules. The same argument applies for the first isomorphism. See [CPS1, Thm. 3.9]. The second isomorphism is a consequence of Thm. 1.1 and the fact, noted above, that j^{∗}Δ(ω) is the standard object in the highest weight category C_{G}(Ω_{F}^{+}). The third isomorphism follows from [CPS6, Lemma 6]; it is a general fact involving a quotient functor j^{∗} : C_{G}[Γ^{+}] → (Ω^{+}) for any arbitrary coideal in a finite weight poset Γ^{+}.

3If G has simple derived group G′, then Ω^{+} is itself finite [CPS6, Prop. 9], so we can usually take F = Ω^{+}. However, even in that case, it might be desirable to consider a proper subset F.

Similarly, there are isomorphisms

Ext_{H}^{•}(π_{Ω}M, ∇_{H}(ω)) ≅ Ext_{C_{H}[Ω_{F}^{+}]}^{•}(\overline{π_{Ω}}j^{∗}M, ∇_{H}(ω))
≅ Ext_{C_{G}(Ω_{F}^{+})}^{•}(j^{∗}M, j^{∗}∇(ω))
≅ Ext_{C_{G}[Γ_{F}^{+}]}^{•}(M, ∇(ω))
≅ Ext_{G}^{•}(M, ∇(ω)).

(1.1.3)

In particular, in both (1.1.2) and (1.1.3), we can take M = L(τ), for τ ∈ Ω_{F}^{+}, to obtain

(1.1.4) Ext_{G}^{•}(Δ(ω), L(τ)) ≅ Ext_{H}^{•}(Δ_{H}(ω), L_{H}(τ))

and

Ext_{G}^{•}(L(τ), ∇(ω)) ≅ Ext_{H}^{•}(L(τ), ∇(ω)). (1.1.5)

See [CPS6, pp. 224–226] for further discussion, including the relation of these Extgroups with Kazhdan-Lusztig polynomials. In §§§6,7, 8, we will take up the issue of comparing Ext-groups between irreducible modules.

2. Some elementary applications

As noted in [CPS6, §6], the proof of Thm. 1.1 relies entirely on properties of highest weight categories, and so it remains valid in other contexts, such as quantum enveloping algebras and q-Schur algebras at a root of unity. In this section, we present various illustrations. These results will not be used elsewhere in this paper.

First, we introduce some standard notation (which will be important also in §7). If λ = (λ_{1}, ⋯, λ_{n}) is a partition of r we write λ ⊢ r; let Λ^{+}(r) be the set of all partitions of r, and Λ^{+}(n, r) the set of partitions with at most n nonzero parts. If λ ⊢ r, write |λ| = r. The set Λ^{+}(r) is regarded as a poset, using the dominance ordering ⊴: λ ⊴ μ ⟺ ∑_{i=1}^{j} λ_{i} ≤ ∑_{i=1}^{j} μ_{i} for all j. Then Λ^{+}(n, r) is a coideal in Λ^{+}(r). For λ ∈ Λ^{+}(r), let λ′ denote the dual partition; thus, λ′_{i} = #{λ_{j} ≥ i}.

Let 𝔼 be a vector space over the field of real numbers ℝ of dimension n. Assume that 𝔼 is a Euclidean space with orthonormal basis ϵ_{1}, ⋯, ϵ_{n}. We consider the root system Φ of type A_{n−1} of 𝔼 which has roots ϵ_{i} − ϵ_{j}, 1 ≤ i ≠ j ≤ n. We label the roots, etc. in the standard way, e. g., as in [Bo, p. 250]. Thus, Φ has a simple set Π = {α_{1}, ⋯, α_{n−1}} of roots, putting α_{i} − ϵ_{i} − ϵ_{i+1}, 1 ≤ i < n. We can identify the ℤlattice generated by ϵ_{1}, ⋯, ϵ_{n} with the character group X = X(T) (written additively) of the n-dimensional torus T = 𝔾_{m}^{×n}, setting ϵ_{i} : T → 𝔾_{m} to be the projection onto the i-th factor. With this notation, the “fundamental dominant weights” are defined by putting ϖ_{i} = ϵ_{1} + ⋯ + ϵ_{i}, i = 1, ⋯, n − 1. Using the ℤ-basis ϵ_{1}, ⋯, ϵ_{n} for X, we can identify X as the abelian group of sequences (λ_{1}, ⋯, λ_{n}) of integers. Then X^{+} identifies with the set of all (λ_{1}, ⋯, λ_{n}) satisfying λ_{1} ≥ ⋯ ≥ λ_{n}. In this way, Λ^{+}(n, r) ⊂ X^{+}.

For 1 ≤ d < n, Π_{d} := Π\{α_{d}} is a base for a subroot system Φ_{d} of type A_{j−1}×A_{n−1−j}. For λ, μ ∈ Λ^{+}(n, r), write λ =_{d} μ provided that λ − μ ∈ ℤΦ_{d}. Also, put λ^{[d]} = (λ_{1}, ⋯, λ_{d}). The following result is immediately verified.

Lemma 2.1. For λ, μ ∈ Λ^{+}(n, r), λ =_{d} μ if and only if |λ^{[d]}| = |μ^{[d]}|.

Now, using Thm. 1.1, we easily obtain expressions for various Ext-groups involving algebraic groups (or quantum groups) of type A. In turn, the answers can be given in terms of Schur algebras S(n, r) (or q-Schur algebras S_{q}(n, r)). We illustrate these results in several cases, referring the reader to [CPS6] for an explanation of the (very standard) notation, and further references. In the first four examples, we work with q-Schur algebras, letting q ∈ k be a primitive ℓth root of unity.

Example 1: Row removal. Suppose λ, μ ∈ Λ^{+}(n, r) satisfy λ_{1} = μ_{1}. Let \overline{λ} = (λ_{2}, ⋯, λ_{n}), \overline{μ} = (μ_{2}, ⋯, μ_{n}) ∈ Λ^{+}(n − 1, r′), where r′ = r − λ_{1}. Then for (M, N) ∈ {(Δ, L), (L, ∇), (Δ, Δ), (∇, ∇)}, we have

Ext_{S_{q}(n,r)}^{•}(M(λ), N(λ)) ≅ Ext_{S_{q}(n−1,r′)}^{•}(M(\overline{λ}), N(\overline{μ})).

In fact, this follows immediately from Thm. 1.1, using Lemma 2.1 with d = 1. This result has already been observed in [CPS6, §§4,6].

Example 2: Removal of several rows. Keep the notation of Example 1 above, but replace Φ_{1} by Φ_{d} for some d satisfying 1 < d < n. Now given λ, μ ∈ Λ^{+}(n, r), suppose that |λ^{[d]}| = |μ^{[d]}| = r′ and set r′′ = r − r′. With (M, N) as before,

Ext_{S_{q}(n,r)}^{•}(M(λ), N(λ)) ≅
Ext_{S_{q}(d,r′)}^{•}(M(λ^{[d]}), N(μ^{[d]})) ⊗ Ext_{S_{q}(n−d,r′′)}^{•}(M(λ\λ^{[d]}), N(μ\μ^{[d]})),

as graded vectors spaces. We have written λ\λ^{[d]} for (λ_{d+1}, ⋯, λ_{n}) and similarly for μ. In checking these facts, it is useful to make use of the fact that, for positive integers a, b, Ext_{S_{q}(a,b)}^{•}(M(λ), N(λ)) ≅ Ext_{S_{q}(b,b)}^{•}(M(λ), N(λ)) for λ, μ ∈ Λ^{+}(a, b). (On the right-hand side, we interpret λ, μ ∈ Λ^{+}(b).) Suitably interpreted in the context of q-Schur algebras, this isomorphism is just that given in (1.1.4) or (1.1.5); see [PS, Thm. 6.1].

Example 3: Passage to Hecke algebras. Define e = \begin{cases}p if ℓ = 1; \\ ℓ otherwise, \end{cases} where k

has characteristic p. Let H_{r} be the Hecke algebra associated to the symmetric group 𝔖_{r} and the parameter q. We use the notation of [PS]. For λ ⊢ r, let S_{λ} denote the corresponding Specht module for H_{r}. Let m be a non-negative integer satisfying m ≤ e − 3. Consider the setting of Example 2 with n = r, and let λ, μ ∈ Λ^{+}(r) satisfy λ^{[d]} = μ^{[d]}. Then

Ext_{H_{r}}^{m}(S_{λ}, S_{μ}) = ⨁_{i=0}^{m} Ext_{H_{s}}^{i}(S_{λ^{[d]}}, S_{μ^{[d]}}) ⊗ Ext_{H_{t}}^{m−i}(S_{λ\λ^{[d]}}, S_{μ\μ^{[d]}}).

This result is immediate from [PS, Thm. 4.6(ii)].

Example 4: Column removal. Fix d, 1 ≤ d < n, and suppose that λ, μ ∈ Λ^{+}(n, r) satisfy λ′ =_{d} μ′. Then we can use Example 2 to give a Künneth factorization of Ext_{S_{q}(n,r)}^{•}(Δ(λ), L(μ)), after observing that

(2.1.1) Ext_{S_{q}(r,r)}^{•}(Δ(λ), Δ(μ)) ≅ Ext_{S_{q}(r,r)}^{•}(Δ(μ′), Δ(λ′)).

To see this, we use some tilting module results, summarized (with references) in [CPS6, §3]. Let Y be a complete tilting module for S_{q}(r, r)–mod. Set E_{q}(r, r) = End_{S_{q}(r,r)}(Y). Form the contravariant tilting functor T : S_{q}(r, r)–mod → E_{q}(r, r)–mod given by T (M) := Hom_{S_{q}(r,r)}(M, Y). The category E_{q}(r, r)–mod is equivalent to the category S_{q}(r, r)–mod, so we identify these two categories. (Caution: the equivalence is not by means of the tilting functor T.) With this identification, T Δ(λ) ≅ Δ(λ′). If P_{•} → Δ(λ) → 0 is a projective resolution, then 0 → Δ(λ′) → T P_{•} is a resolution of Δ(λ′) ≅ T Δ(λ) by tilting modules; see [PS, Lemmas 1.4&1.5]. Since tilting modules are acyclic for the functor Hom_{E_{q}(r,r)}(Δ(μ′), −), (2.1.1) follows from the isomorphism

Hom_{E_{q}(r,r)}(Δ(μ′), T P_{•}) ≅ Hom_{S_{q}(r,r)}(P_{•}, Hom_{E_{q}(r,r)}(Δ(μ′), Y))

(see [PS, Prop. 1.2]) and the fact that Hom_{E_{q}(r,r)}(Δ(μ′), Y) ≅ Δ(μ). More generally, this argument shows that T defines a contravariant equivalence from the exact subcategory of S_{q}(r, r)-modules with a Δ-filtration to the similar category for E_{q}(r, r)–mod.

In Example 2, the isomorphism

Ext_{S_{q}(n,r)}^{•}(∇(λ), ∇(μ)) ≅ Ext_{S_{q}(d,r′)}^{•}(∇_{S_{q}(d,r′)}(λ), ∇_{S_{q}(d,r′)}(μ))

is already essentially proved in [D2, Formula (17), p. 91], without the explicit interpretation in terms of the decomposition as a tensor product.

The Künneth results above have only been worked out in type A. More generally, in other types in the case of a reductive algebraic (or quantum) group G, a similar such tensor factorization would exist whenever the semisimple part of the Levi subgroup H decomposes into a product of smaller reductive groups. However, it is not likely that the elegant interpretation in type A given above holds in all other cases. Our last example below indicates that this does occur in type C_{n} at least.

Example 5: Type C. Suppose that G ≅ Sp_{2n}(k) has type C_{n}. We use the notation of [Bo, p.254-255] (replacing l there by n) as far as listing the set Π of simple roots, etc. Thus, X = X(T) ≅ ℤϵ_{1} ⊕ ⋯ ⊕ ℤϵ_{n}, and the fundamental dominant weights ϖ_{i}, 1 ≤ i ≤ n, are given by ϖ = ϵ_{1} + ⋯ + ϵ_{i}. The set Λ^{+}(n, •) of all partitions λ of length at most n indexes the dominant weights X^{+}. Just as for type A, for 1 ≤ d ≤ n, define Φ_{d} as the subroot system of type A_{d−1} × C_{n−d−1} of Φ with simple roots Π_{d} := Π\{α_{d}}. Put λ =_{d} μ if and only if λ − μ ∈ ℤΦ_{d}. Then, by analogy with Lemma 2.1, given λ, μ ∈ Λ^{+}(n, •), we verify that λ =_{d} μ if and only if |λ^{[d]}| = |μ^{[d]}| and |λ| ≡ |μ| mod 2. If H is the Levi subgroup of G corresponding to Φ_{d}, then H ≅ GL_{d}(k) × Sp_{2(n−1−d)}(k).

Thus, if λ =_{d} μ, the groups Ext_{G}^{•}(M(λ), N(λ)) are isomorphic as a graded vector space to

Ext_{GL_{d}(k)}^{•}(M(λ^{[d]}), N(μ^{[d]})) ⊗ Ext_{Sp_{2g}(k)}^{•}(M(λ\λ^{[d]}), N(μ\μ^{[d]})),

where g = n − d − 1 and (M, N) ∈ {(Δ, L), (L, ∇), (Δ, Δ), (∇, ∇)}. We leave further details to the interested reader.

PART II: Character formulas

We study various conditions equivalent to the validity of the Lusztig character formula. We arrange our formulations, in this section and the next, so that we can discuss cases where the formula might hold, even though the full Lusztig conjecture might not hold, or might not be known to hold.

3. The Lusztig character formula

Let G be a simple and simply connected algebraic group over k. (The results below and in Part III easily extend to the case of a general reductive group.) In addition, let ρ ∈ X^{+} be defined by ⟨ρ, α^{∨}⟩ = 1 for all simple roots α ∈ Π. Let W = N(T)/T be the Weyl group of G and let W_{p} := W ⋉ pℤΦ be the affine Weyl group, generated by W and the normal subgroup pℤΦ of translations by p-multiples of roots. It acts naturally (and faithfully) on the space 𝔼 = ℝ ⊗ X. For α ∈ Φ, n ∈ ℤ, s_{α,np} ∈ W_{p} is defined as an operator on 𝔼 by s_{α,np}(u) = u − (⟨u, α^{∨}⟩ − np)α, u ∈ 𝔼. Set S_{p} = {s_{α} | α ∈ Π} ∪ {s_{α_{o},−p}} ⊂ W_{p}, where α_{o} ∈ Φ is the maximal short root. Then (W_{p}, S_{p}) is a Coxeter system; let l : W_{p} → ℤ be the length function of W_{p} defined by S_{p}. In place of the natural action of W_{p} on 𝔼, we prefer the “dot” action given by w ·u = w(u+ρ)−ρ, w ∈ W_{p}, u ∈ 𝔼. Given α ∈ Φ, n ∈ ℤ, let H_{α,np} = {u ∈ 𝔼 | ⟨u + ρ, α^{∨}⟩ = pn}. Thus, s_{α,np} ∈ W_{p} is a reflection about the hyperplane H_{α,np}. Any connected component C of 𝔼\ ⋃_{α∈Φ,n∈ℤ} H_{α,np} is an alcove of W_{p}. The closure \overline{C} a fundamental domain for the dot action of W_{p} on 𝔼.

A weight λ ∈ X is called regular provided that λ belongs to some alcove C of W_{p}, i. e., provided ⟨λ + ρ, α^{∨}⟩ ≢ 0 mod p for all α ∈ Φ. If h denotes the Coxeter number of G, regular weights exist if and only if p ≥ h. In particular, if p ≥ h, −2ρ = w_{o} · 0 is a regular weight, where w_{o} denotes the long word in W. If Σ ⊂ 𝔼, let Σ_{reg} be the set of all regular elements in Σ.

For the rest of this section, assume that p ≥ h. By definition, the Lusztig character formula holds for λ ∈ X^{+} (in the category C_{G} of finite dimensional rational G-modules) provided λ = x · (−2ρ) for some x ∈ W_{p} and the irreducible G-module L(λ) has formal character given by

ch L(λ) = ∑_{y·(−2ρ)∈X^{+}}(−1)^{l(x)−l(y)}P_{y,x}(1)ch Δ(y · (−2ρ)). (3.0.2)

In this formula, P_{y,x} is the Kazhdan-Lusztig polynomial^{4} associated to the pair (y, x) ∈ W_{p} × W_{p}. In particular, P_{y,x} = 0 unless y ≤ w in the Bruhat-Chevalley ordering on W_{p}. For any ξ ∈ X^{+},

ch Δ(ξ) = \frac{∑_{w∈W} e^{w·ξ}}{∑_{w∈W} e^{w·0}}

by Weyl’s character formula. Given a subset Σ ⊆ W_{p} · 0 = W_{p} · (−2ρ), we say that the Lusztig character formula holds for Σ it if holds for every λ ∈ Σ.

The famous Lusztig conjecture for modular characters [L] states that the Lusztig character formula holds for W_{p} · 0 ∩ Γ_{Jan}, where

Γ_{Jan} = {λ ∈ X^{+} | ⟨λ + ρ, α^{∨}⟩ ≤ p(p − h + 2), ∀α ∈ Φ} (3.0.3)

is the Jantzen region. Of course, to test if λ ∈ Γ_{Jan}, it suffices to check the inequality in (3.0.3) just for α = α_{o}. For a group G with a fixed root system Φ, the conjecture holds provided p is large enough, through, in general, no sufficient bound on p is known. See [AJS] and, for a general discussion, [T, §8]. The Lusztig conjecture holds generally in types A_{n}, n ≤ 3, and B_{2}, G_{2} [Jan]. In addition, it is known to hold in type A_{4} for p = 5, 7; see the discussion in [S2].

However, independently of the validity of the Lusztig conjecture, ideals Γ for which the Lusztig character formula holds for W_{p} · 0 ∩ Γ occur naturally, as Thm. 4.1 below suggests. A simple (almost trivial) example would be to take Γ = {λ ∈ X^{+} | λ ≤ s_{α_{o},p} · 0}.

Let C^{+} ⊂ 𝔼 be the alcove containing 0, and put C^{−} = w_{o} ·C^{+} (the alcove containing −2ρ). Let λ ∈ X ∩ C^{−} and suppose the Lusztig character formula holds for x · (−2ρ), x ∈ W_{p}. Using translation,

ch L(x · λ) = ∑_{y·λ∈X^{+}}(−1)^{l(x)−l(y)}P_{y,x}(1)ch Δ(y · λ). (3.0.4)

Thus, we can say that the Lusztig character formula holds for x · λ. Suppose that Σ ⊂ X^{+} has the property that Σ_{reg} is a union of sets of the form C ∩ X^{+}, C an alcove. (For example, Γ_{Jan} has this property.) Then, if the Lusztig character formula holds for W_{p} · 0 ∩ Σ, it also holds for Σ_{reg}.^{5}

Let X_{1}^{+} be the set of p-restricted dominant weights. A conjecture of Kato [K, Conj. 5.5] asserts that the Lusztig character formula holds for the set X_{1}^{+} ∩ W_{p} · 0 if p ≥ h.

4The polynomials P_{y,x} are associated to the generic Hecke algebra H for the Coxeter system (W_{p}, S_{p}). Each P_{y,x} is, in fact, a polynomial in u := t^{2} and P_{x,y}(1) means the specialization at u = 1 (i. e., at t = −1). The introduction of the variable t = \sqrt{u} is only important in this paper in the proof of Thm. 4.1(b),(c) below, involving the theory of enriched Grothendieck groups.

5Suppose λ′ ∈ X ∩ \overline{C^{−}} is not regular. Let D_{λ′} be set of distinguished left coset representatives in W_{p} of the stabilizer in W_{p} of λ′ under the dot action. Suppose that (3.0.4) holds for a given x ∈ D_{λ}.

Then, using Jantzen translation, ch L(x · λ′) = ∑_{y·λ′∈X^{+}}(−1)^{l(x)−l(y)}P_{y,x}(1)ch Δ(y · λ′). In this way, it would be possible to speak of the Lusztig character formula holding for an ideal Γ, not just Γ_{reg}, though we will not use this more general terminology.

Observe that X_{1}^{+} ⊆ Γ_{Jan} if and only if p ≥ 2h − 3. If his conjecture holds, then (the argument in) [K, (5.4)] also establishes that (3.0.2) holds for any L(λ) = L(τ)⊗L(ξ)^{(1)}, where τ ∈ X_{1 reg}^{+} and ξ ∈ X^{+} has the property that Δ(ξ) ≅ L(ξ).^{6} Here, given a rational G-module V, V^{(1)} denotes the “twist” of V through the Frobenius morphism Fr : G → G.

The following result will be useful in §7. Although the statement of (a) is purely combinatorial, the authors are unaware of a purely combinatorial proof. It would also be interesting to remove the various restrictions in the statement this result.

Lemma 3.1. Assume that p ≥ h and that the root system of G is simply laced. In addition, if G has type E_{6}, E_{7}, E_{8} assume that p > h + 1. Suppose that x ∈ W_{p} and x · (−2ρ) = pλ for λ ∈ X^{+}.

(a) We have

ch Δ(λ)^{(1)} = ∑_{y·(−2ρ)∈X^{+}}(−1)^{l(x)−l(y)}P_{y,x}(1)ch Δ(y · (−2ρ)).

(b) If Δ(λ) ≠ L(λ), the Lusztig character formula (3.0.2) does not hold for pλ = x · (−2ρ).

Proof. Let U_{q} be the quantum enveloping algebra (integral form) at q = \sqrt[p]{1} of the same type as G. Let C_{q} be the category of finite dimensional integral type 1 modules for U_{q}. It is a highest weight category with irreducible (resp. standard) modules L_{q}(ξ) (resp., Δ_{q}(ξ)), ξ ∈ X^{+}. Furthermore, for any ξ = x · (−2ρ) ∈ X^{+}, the Lusztig character formula holds for ξ in the category C_{q}; in other words, (3.0.2) remains valid if each “L” (resp., “Δ”) is replaced by “L_{q}” (resp., “Δ_{q}”) This last assertion follows essentially from the validity of Lusztig’s quantum group conjecture: see [T, §7] for a detailed discussion and further references. It is here that the restriction that p > h + 1 is required in the exceptional cases—see the (equivalent) case–by–case formulation in [T, Thm. 7.1]. Of course, ch Δ_{q}(ξ) = ch Δ(ξ) for any ξ ∈ X^{+}.

Let σ : U_{q} → U be the Frobenius morphism from U_{q} to the universal enveloping algebra over ℂ of the complex Lie algebra 𝔤 having the same type as G. Then Δ_{q}(pλ) = σ^{∗}L_{ℂ}(λ), the pull-back through σ of the complex irreducible 𝔤-module of highest weight λ. Hence, ch Δ_{q}(pλ) = ch Δ(λ)^{(1)}, and (a) follows.

Since L(pλ) ≅ L(λ)^{(1)}, (b) also follows. □

Remark 3.2. See [T] for a discussion of the status of the Lusztig quantum group conjecture used in the above proof in the non-simply laced case. Published results giving a character formula require p ≥ r_{0}, where r_{0} is sufficiently large, depending on the root system, but unknown; see [T, Thm. 7.1] and remarks following it. However,

[T] also notes an improvement to p > h in all cases using unpublished work of Soergel. A similar improvement in all cases might be based on the category equivalences given in [ABG], though specific details leading from the final result in [ABG] to a character formula are not included. In our paper, the above lemma is only used in type A. In fact, in this case, as noted in [T], the equivalence between representations of the quantum enveloping algebra at an ℓth root of unity and the corresponding representations for affine Lie algebras holds for all ℓ, without any ℓ ≥ h restriction.

6In general, such weights τ + pξ need not always lie in Γ_{Jan}.

4. Equivalent conditions

Assume G is simple (any type), simply connected and p ≥ h. The faces of an alcove C are naturally labelled by elements of S_{p}: if C = C^{−}, the faces are already labelled by S_{p} by definition; otherwise, a face F is W_{p} dot-conjugate to a unique face of C^{−}, and we assign to F the corresponding element s ∈ S_{p}. Thus, for s ∈ S_{p}, we speak of an s-face of C. For λ, μ ∈ X_{reg}^{+} and s ∈ S_{p}, write μ = λs provided: (1) λ, μ lie in adjacent alcoves C, C′ separated by an s-face, and (2) λ is the reflection of μ through that s-face. In other words, λ = s_{β,np}·μ, where the common s-face of C, C′ lies in the hyperplane H_{β,np}. Then λ and λs are called adjacent. If λ = w · (−2ρ), w ∈ W_{p}, then λs = ws · (−2ρ). Also, λs > λ ⟺ l(ws) = l(w) + 1, and λ = w · (−2ρ) ∈ X^{+} ⟺ w = w_{o}y, with y a “distinguished” right coset representative of W in W_{p}, i. e., l(w) = l(w_{o}) + l(y). In particular, if y is distinguished and y = s_{1} ⋯ s_{n}, s_{i} ∈ S_{p}, then each product s_{1} ⋯ s_{m}, m ≤ n, is also a distinguished distinguished right coset representative. Thus,

0 = w_{o} · (−2ρ) < w_{o}s_{1} · (−2ρ) < ⋯ < w_{o}s_{1} ⋯ s_{n} · (−2ρ)

provides a “path” of adjacent dominant weights from 0 to λ.

In the following theorem, part (a) plays a basic role in the sequel. Part (b) gives a way to construct ideals Γ for which the Lusztig character formula holds. However, it and part (c) will not be used in the paper,^{7} so that the proofs are rather brief, and rely heavily on some of the machinery developed in [CPS2], [CPS4].

Theorem 4.1. Let Γ be a finite ideal in the poset (X^{+}, ≤).

(a) The Lusztig character formula holds for Γ ∩ W_{p} · 0 if and only if

Ext_{G}^{1}(L(λ), L(λs)) ≠ 0 (4.1.1)

whenever λ < λs both belong to Γ ∩ W_{p} · 0.

(b) Suppose that there exists τ ∈ X^{+} ∩ W_{p} · 0 such that Γ = {ξ ∈ X^{+} | ξ ≤ τ } and form the ideal Γ̃ = {ξ ∈ Γ | ξ < τ} of Γ. Suppose that s ∈ S_{p} and τs ∈ Γ̃. Assume that the Lusztig character formula holds for Γ̃ ∩ W_{p} · 0. Then the Lusztig character formula holds for Γ ∩ W_{p} · 0 if and only if Ext_{G}^{1}(L(τs), L(τ)) ≠ 0.^{8}

(c) Given ξ ∈ X^{+} ∩W_{p} ·0, the Lusztig character formula holds for ξ provided there is a path 0 = ξ_{0} < ξ_{1} < ⋯ < ξ_{m} = ξ in X^{+} ∩W_{p} ·0 with ξ_{_{i}−1} adjacent to ξ_{i} for 0 < i ≤ m and with each Ext_{G}^{1}(L(ξ_{i−1}), L(ξ_{i})) ≠ 0.

7More precisely, (c) makes a brief cameo appearance in §7.

8In the statement of (b) and (c), we could replace τ and ξ be regular weights, using a standard translation functor argument.

Proof. Part (a) is proved in [CPS2, Thm. 5.3] for Γ = Γ_{Jan}, but the argument holds equally well for any ideal Γ in X^{+}, making use of the discussion in the paragraph immediately above the statement of the theorem.

Although (c) implies (b), we prove (b) first, then augment the argument to obtain (c). In in the proof of (b), we will need to use the theory (and notation) of pre-Hecke operators discussed in [CPS2, §5] and [CPS4, §4] (the latter reference contains many more details). Let C_{G,0} be the category of all finite dimensional rational G-modules with composition factors L(λ) for some λ ∈ W_{p} · 0. Then C_{G,0} is a highest weight category with poset (W_{p} · 0, ↑), letting ↑ be the Jantzen “up arrow” poset structure. Form the exact, additive functor Θ_{s} : C_{G,0} → C_{G,0} (the composite of two Janzten translation functors) [CPS4, p. 88]. If M has the form Δ(λ), ∇(λ), or L(λ) for some λ ∈ W_{p} · 0, then let β_{s}M equal to the complex

0 → M → β_{s}M → M → 0,

defined in [CPS4, (4.8.2)]. It is concentrated in degrees −1, 0, and +1. When λ < λs, β_{s}L(λ) is isomorphic in the bounded derived category D^{b}(C_{G,0}) to its cohomology, which is concentrated in degree 0. (We will take λs = τ in the notation of (b).) From the discussion in [CPS4, pp. 89–90], if λ ∈ Γ̃, β_{s}L(λ) ∈ E^{L} ∩ E^{R}. See also [CPS4, Remark 2.2]. Here E^{L} ∩ E^{R} is a subset of objects in D^{b}(C_{G,0}) defined by an evenodd vanishing property similar to that described for irreducible modules in (8.0.2) below. The condition Ext_{G}^{1}(L(τs), L(τ)) ≠ 0 implies that L(τ) is a direct summand of β_{s}L(τs); see the end of the proof of “(c) ⟹ (b)” of [CPS4, Thm. 5.5]. Therefore, L(τ) ∈ E^{L} ∩ E^{R}. This implies that C_{0}[Γ] has a Kazhdan-Lusztig theory (again, see (8.0.2) below), and so the Lusztig character formula holds for Γ, as required.

The proof of (c) requires the use of the “enriched Grothendieck groups” K_{0}^{L} := K_{0}^{L}(C_{G,0}, l) and K_{0}^{R} := K_{0}^{R}(C_{G,0}, l) introduced in [CPS2, §2] (see also [CPS4, §2]). In fact, these groups can be associated to any highest weight category C, with weight poset Λ and length function l : Λ → ℤ. As discussed in [CPS2, Prop. 2.3], K_{0}^{L} and K_{0}^{R} are related by a non-degenerate sesquilinear pairing K_{0}^{L} × K_{0}^{R} → ℤ[u, u^{−1}] (Laurent polynomials). When C_{G,0}[Γ ∩ W_{p} · 0] has a Kazhdan-Lusztig theory [CPS2, Defn. 3.3] (see also (8.0.2) below), the group K_{0}^{L} (resp., K_{0}^{R}) keeps track of the dimensions of the groups Ext_{C_{G,0}}^{•}(L(λ), ∇(μ)) (resp., Ext_{C_{G,0}}^{•}(Δ(μ), L(λ))) for λ, μ ∈ Γ ∩ W_{p} · 0.

Let H denote the generic Hecke algebra of the Coxeter system (W_{p}, S_{p}) over ℤ[t, t^{−1}]. Then H acts on the groups K_{0}^{L} and K_{0}^{R}, using the β_{s}, s ∈ S_{p}, mentioned above [CPS2, Prop. 5.6]. The argument for (b) shows, recursively, that all the L(ξ_{i})’s may be represented in both K_{0}^{L} and K_{0}^{R}. Also, the discussion in [CPS4, §4] shows the resulting representations are completely determined by Kazhdan-Lusztig polynomials. Using the “Euler characteristic map” K_{L}^{0} → K_{0}(C_{G,0}) (the Grothendieck group of G–mod) obtained by specializing u ↦ −1 (and hence t ↦ 1; cf. footnote 4) yields (3.0.2) for L(ξ). □

Let λ ∈ X_{reg}^{+} and suppose that λ < λs ∈ X^{+} for some s ∈ S_{p}. By [Jan, II, Prop. 7.21], dim Ext_{G}^{1}(Δ(λ), Δ(λs) = 1. In simple terms, this means there is a non-split short exact sequence

0 → Δ(λs) → E → Δ(λ) → 0 (4.1.2)

of rational G-modules, and that, given any other non-split short exact sequence 0 → Δ(λs) → E′ → Δ(λ) → 0, it is “scalar equivalent” to (4.1.2) in the sense that there is a commutative diagram

[figure: 0; Δ(λs); E; Δ(λ); 0]
[figure: α id; id]
[figure: 0; Δ(λs); E′; Δ(λ); 0]

(4.1.3)

in which 0 ≠ α ∈ k. In general,

dim Ext_{G}^{1}(L(λ), L(λs)) ≤ 1.

If a non-trivial extension of L(λ) by L(λs) exists, it must come, up to scalar equivalence, from (4.1.2) by factoring out rad(Δ(λs)). That is, the resulting extension of Δ(λ) by L(λs) must split on rad(Δ(λ)). If it does, we get a non-split extension of L(λ) by L(λs); if it does not, there is no such nontrivial extension. Observe that

(4.1.4) Ext_{G}^{1}(Δ(λ), Δ(λs)) ≅ Ext_{G}^{1}(Δ(λ), L(λs)).

These statements can be easily verified, using the long exact sequence of Ext^{•}. Note that Ext_{G}^{n}(Δ(λ), L(ν)) = 0 unless ν > λ.

For each μ ∈ Γ∩W_{p}·0, let P_{μ} be a fixed projective in the category C_{G}[Γ] such that the projective cover P (μ) of L(μ) is a direct summand of P_{μ}, and all other indecomposable summands P (ν) satisfy μ < ν ∈ Γ.^{9} Explicitly, P_{μ} ∈ C_{G}[Γ], while P (μ) ∈ C_{G}[Γ∩W_{p} ·0], the category of all finite dimensional rational G-modules with composition factors of the form L(ν), ν ∈ Γ ∩ W_{p} · 0. For γ ∈ Γ, let P_{μ,<γ} be the largest quotient module of P_{μ} with all composition factors L(ν), ν < γ.

The module P_{μ,<γ} has a filtration by submodules with successive sections standard modules Δ(τ), τ < γ.

Theorem 4.2. Let Γ be a finite ideal as above. For any given λ, λs ∈ Γ ∩ W_{p} · 0 with λ < λs, s ∈ S_{p}, condition (4.1.1) holds if and only if, for each μ < λ in Γ ∩ W_{p} · 0, the following condition holds:

9Such projective modules P_{μ} arise naturally in many contexts, where P (μ) itself may not be explicitly known. We allow summands P (ν) for ν ∉ W_{p} · 0, though they will be irrelevant, since then L(ν) and L(μ) lie in different blocks.

We only use that the other summands P (ν) of P_{μ} satisfy μ ≤ ν in this paper, though the natural examples have μ < ν. That is, we assume that P (μ) is a summand of P_{μ} with multiplicity one.

For any map P_{μ,<λs} → Δ(λ) which is not a split surjection, the induced map Ext_{G}^{1}(Δ(λ), Δ(λs)) → Ext_{G}^{1}(P_{μ,<λs}, Δ(λs)) is zero.

(4.2.1)

Thus, the Lusztig character formula holds for Γ ∩ W_{p} · 0 if and only if (4.2.1) holds for all μ, λ, λs ∈ Γ ∩ W_{p} · 0.

Proof. We will make use of elementary recollement results; these are described in [CPS6, §2], for example. Let Γ′ be an ideal in Γ. If M ∈ C_{G}[Γ], M_{Γ′} denotes the largest quotient module of M all of whose composition factors lie in Γ′. If P ∈ C_{G}[Γ] is projective, then P_{Γ′} ∈ C_{G}[Γ′] is projective. Moreover, the natural inclusion C_{G}[Γ′] ⊆ C_{G}[Γ] induces a full embedding D^{b}(C_{G}[Γ′]) → D^{b}(C_{G}[Γ]) of bounded derived categories. In particular, given M, N ∈ C_{G}[Γ′],

Ext_{C_{G}[Γ′]}^{•}(M, N) ≅ Ext_{C_{G}[Γ]}^{•}(M, N) ≅ Ext_{G}^{•}(M, N).

It follows that Ext_{G}^{1}(P_{μ,<λs}, rad(Δ(λs))) = 0 and Ext_{G}^{2}(P_{μ,<λs}, radΔ(λs)) = 0, so that Ext_{G}^{1}(P_{μ,<λs}, Δ(λs)) ≅ Ext_{G}^{1}(P_{μ,<λs}, L(λs)). Similarly, we have the result (4.1.4) already noted. So, for any map P_{μ,<λs} → Δ(λ), there is a commutative diagram

Ext_{G}^{1}(Δ(λ), Δ(λs))  \overline{}Ext_{G}^{1}(P_{μ,<λs}, Δ(λs))

[figure]
Ext_{G}^{1}(Δ(λ), L(λs))  \overline{}Ext_{G}^{1}(P_{μ,<λs}, L(λs))

(4.2.2)

in which the vertical maps are isomorphisms.

Now suppose that P_{μ,<λs} →^{ϕ} Δ(λ) is not a split surjection. Then ϕ is not surjective, since Δ(λ) ∈ C_{G}[Γ′] is projective. So ϕ factors as P_{μ,<λs} → rad(Δ(λ)) ⊆ Δ(λ).

Now assume that (4.1.1) holds. Then the canonical nonzero extension of Δ(λ) by L(λs) is zero on rad(Δ(λ)), so the map Ext_{G}^{1}(Δ(λ), L(λs)) → Ext_{G}^{1}(P_{μ,<λs}, L(λs)) must be zero, as is the map Ext_{G}^{1}(Δ(λ), Δ(λs)) → Ext_{G}^{1}(P_{μ,<λs}, Δ(λs)). Thus, (4.1.1) implies (4.2.1).

Conversely, suppose condition (4.2.1) holds. Let P_{<λs} be a direct sum of sufficiently many P_{μ,<λs} with μ < λ, μ ∈ Γ∩W_{p} ·0, so that there is a surjection P_{<λs} → rad(Δ(λ)). Obviously, the component maps P_{μ,<λs} → P_{<λs} → rad(Δ(λ)) do not give split surjections when composed with the inclusion rad(Δ(λ)) ⊆ Δ(λ). Thus, the composites Ext_{G}^{1}(Δ(λ), Δ(λs)) → Ext_{G}^{1}(P_{μ,<λs}, Δ(λs)) are zero by (4.2.1). Using the above diagram, we obtain that the map

Ext_{G}^{1}(Δ(λ), Δ(λs)) → Ext_{G}^{1}(rad(Δ(λ)), L(μ)) → Ext_{G}^{1}(P_{μ,<λs}, L(λs))

is zero for each component P_{μ,<λs} of P_{<λs}. Thus, the map

Ext_{G}^{1}(Δ(λ), L(λs)) → Ext_{G}^{1}(rad(Δ(λ)), L(λs)) → Ext_{G}^{1}(P_{<λs}, L(λs))

is also zero. However, the long exact sequence of Ext shows that that the map Ext_{G}^{1}(rad(Δ(λ)), L(λs)) → Ext_{G}^{1}(P_{<λs}, L(λs)) is injective, since Hom_{G}(N, L(λs)) for the kernel N of P_{<λs} → rad(Δ(λ)). This proves that the validity of (4.2.1) for all possible μ, λ, λs as above, implies the validity of (4.1.1). □

The following results ties the above equivalence in with the even–odd varnishing equivalence (as a sufficient condition) for the Lusztig conjecture, see, e. g. [PS, §7]. See also §8 below.

Proposition 4.3. Let Γ be a finite ideal in X^{+}. Fix λ, λs ∈ Γ with s ∈ S_{p}, such that λ < λs. Suppose that condition (4.2.1) fails for some μ < λ. If μ < λ is chosen maximal for which this failure occurs, then Ext_{G}^{1}(Δ(μ), L(λ)) ≠ 0 and Ext_{G}^{1}(Δ(μ), L(λs)) ≠ 0.

In particular, condition (4.2.1) holds if Ext_{G}^{1}(Δ(μ), L(γ)) = 0 always whenever γ ∈ Γ∩W_{p}·0 and μ, γ have lengths of the same parity with respect to some “length” function l : Γ ∩ W_{p} · 0 → ℤ for which l(λs) ≡ l(λ) + 1 mod 2.

Proof. If (4.2.1) fails for some μ, then the argument for Thm. 4.2 shows that the homomorphism ψ : Ext_{G}^{1}(Δ(λ), L(λs)) → Ext_{G}^{1}(rad(Δ(λ)), L(λs)) is not 0. Moreover, if E is a nonsplit extension 0 → L(λs) → E → Δ(λ) → 0, and if F is the largest submodule of E without L(λs) as a composition factor, then ψ factors through Ext_{G}^{1}(E′, L(λs)), where F′ is the image of F in Δ(λ) and E′ = Δ(λ))/F′. Also, (4.2.1) fails for μ if and only if L(μ) is a composition factor of E′ = rad(Δ(λ))/F′. Taking μ maximal among such weights, there are nonzero homomorphisms Δ(μ) → E′ and Δ(μ) → E^{′∗}, where E^{′∗} is the dual of E′. (As well-known, the category C_{G}[Γ] has a duality M ↦ M^{∗} which fixes irreducible modules.) Pulling back to Δ(μ) the evident extensions of E′ (resp., E^{′∗}) by L(λs) (resp., L(λ)) gives the required nonsplit extensions of Δ(μ) by L(λs) (resp., L(λ)). □

The papers [CPS2], [CPS3] contain many other conditions which are equivalent to the validity of the Lusztig character formula. We will return to the even-odd vanishing result mentioned in the proof of Thm. 4.1 later in §8.

5. Type A

We now consider the case G = SL_{n}(k). For technical reasons, we will assume throughout this section that p > 3. See the discussion below concerning (5.0.3).

For positive integers n, r, let (as before) Λ^{+}(n, r) be the set of all partitions of r of length at most n. We regard Λ^{+}(n, r) as a poset, using the dominance ordering ⊴. We will need to make use of a well-known method of associating to any partition with at most n nonzero parts a dominant weight in X^{+}. For this, we will use a local notational convention (i. e., it will be used only in this section!): We will denote partitions by symbols \underline{λ}, \underline{μ}, etc. Thus, if \underline{λ} ∈ Λ^{+}(n, r), then \underline{λ} = (\underline{λ}_{1}, \underline{λ}_{2}, ⋯, \underline{λ}_{n}). Label the fundamental dominant weights ϖ_{1}, ⋯, ϖ_{n−1} for SL_{n}(k) as in [Bo, p. 250]. We have a mapping Λ^{+}(n, r) → X^{+} which associates to \underline{λ} ∈ Λ^{+}(n, r) the dominant weight λ = ∑_{i=1}^{n−1} a_{i}ϖ_{i}, where a_{i} = \underline{λ}_{i} − \underline{λ}_{i+1}.

If a partition \underline{λ} ∈ Λ^{+}(n, r) corresponds to a dominant weight λ ∈ X^{+} as above, then λ is regular (in the sense of §3) if and only if

\underline{λ}_{i} − \underline{λ}_{j} ≢ i − j mod p, ∀i < j ≤ n (5.0.1)

(This alcove geometry notion of regularity is different than the Young diagram notion of p-regularity, which requires that no row be repeated p times.)

For a fixed pair (n, r) of positive integers, let S(n, r) be the Schur algebra over k of bidegree (n, r). Our notation will largely be consistent with that in [CPS6, §4], where the reader can find for more details on Schur algebras in the spirit of this paper. (Also, in the notation of §2 in Part I, S(n, r) = S_{q}(n, r) for q = 1.) The category S(n, r)– mod identifies with the category of rational GL_{n}(k)-modules which are polynomial of homogeneous degree r. The category S(n, r)–mod is a highest weight category with poset (Λ^{+}(n, r), ⊴).

Now fix a finite ideal Γ ⊂ X^{+}. Choose a sufficiently large r ≡ 0 mod n such that every weight in Γ ∩ W_{p} · 0 corresponds (as described above) to a partition \underline{λ} ∈ Λ^{+}(n, r). Any S(n, r)-module M is naturally a GL_{n}(k)-module and hence an SL_{n}(k)-module, and C_{G}[Γ ∩ W_{p} · 0] is a full subcategory of the image of S(n, r)–mod → SL_{n}(k)–mod under this identification functor, with standard, costandard, and irreducible modules going to standard, costandard, and irreducible modules, respectively. In addition, if M, N ∈ S(n, r)–mod, then

(5.0.2) Ext_{S(n,r)}^{•}(M, N) ≅ Ext_{GL_{n}(k)}^{•}(M, N) ≅ Ext_{SL_{n}(k)}^{•}(M, N),

The first isomorphism is well-known (see [CPS6, (6)] and the references there) and the second one follows immediately from an elementary Hochschild-Serre spectral sequence argument, using the normal subgroup G = SL_{n}(k) of GL_{n}(k).

Let mod–k𝔖_{r} denote the category of all finite dimensional modules for the symmetric group 𝔖_{r} of degree r over k. We will make use of Schur-Weyl duality between S(n, r)– mod and mod–k𝔖_{r}. In particular, let S(n, r)–mod(Δ) be the full, exact subcategory of finite dimensional S(n, r)-modules which have a Δ-filtration, i. e., a filtration with sections of the form Δ(\underline{λ}), \underline{λ} ∈ Λ^{+}(n, r). Similarly, let mod–𝔖_{r}(S_{n,r}) be the full, exact subcategory of all finite dimensional (right) k𝔖_{r}-modules which have a filtration with sections of the form S\underline{λ}, \underline{λ} ∈ Λ^{+}(n, r), where S\underline{λ} is the Specht module for k𝔖_{r} corresponding to the partition \underline{λ}. Now assume that p ≥ n = h and recall the blanket assumption that p > 3. By [CPS6, (3.8.3.2)],^{10} there is a contravariant equivalence

(5.0.3) S(n, r)–mod(Δ) ⟶_{∼}^{contra} mod–k𝔖_{r}(S_{n,r}).

If \underline{μ} ∈ Λ^{+}(n, r), let 𝔖\underline{μ} be the associated Young (or parabolic) subgroup of 𝔖_{r}. Let T\underline{μ} = Ind_{𝔖\underline{μ}}^{𝔖_{r}}k be the corresponding permutation module obtained by inducing the trivial module k from 𝔖\underline{μ} to 𝔖_{r}. If V is the standard n-dimensional module of column vectors for GL_{n}(k), then 𝔖_{r} acts on tensor space V^{⊗r} by place permutation. As a right k𝔖_{r}- module, V^{⊗r} decomposes into a direct sum of permutation modules T\underline{λ}, \underline{λ} ∈ Λ^{+}(n, r) (each summand occurring with some positive multiplicity). Then Hom_{k𝔖_{r}}(V^{⊗r}, V^{⊗r}) ≅ S(n, r). The contravariant equivalence (5.0.3) is given explicitly by restricting to the exact subcategory S(n, r)–mod(Δ) the functor

S(n, r)–mod → mod–k𝔖_{r},

M ↦ M^{⋄} := Hom_{S(n,r)}(M, V^{⊗r}). (5.0.4)

The equivalence (5.0.3) takes the standard module Δ(\underline{λ}) ∈ S(n, r)–mod to the Specht module S\underline{λ}. Also, the permutation module T\underline{λ} corresponds to a projective module P\underline{λ} := Hom_{k𝔖_{r}}(T\underline{λ}, V^{⊗r}) of the type discussed in §4, viz., it is a direct sum of the projective indecomposable module P (\underline{λ}) (with multiplicity one), together with various P (\underline{τ}), for \underline{τ} ▷ \underline{λ}, letting ⊵ denote the dominance order on Λ^{+}(n, r). For more details, see [CPS5], [DPS1], and [DPS2].

As a consequence of (5.0.3) and the uniqueness of the short exact sequence (4.1.2), we have the following result. (Recall that p > 3 throughout this section.)

Lemma 5.1. Let Γ be a finite ideal in X^{+} as above. Choose r ≡ 0 mod n so that every λ ∈ Γ ∩ W_{p} · 0 corresponds to a partition \underline{λ} ∈ Λ^{+}(n, r). Suppose λ < λs both belong to Γ ∩ W_{p} · 0 (where s ∈ S_{p}). Then, up to scalar equivalence, there is a unique non-split short exact sequence

0 → S\underline{λ} → F → S\underline{λs} → 0 (5.1.1)

in k𝔖_{r}–mod.

Each weight ν ∈ Γ∩W_{p} ·0 is represented by a partition \underline{ν} ∈ Λ^{+}(n, r). The weight 0 is represented \underline{0} = (r/n, ⋯, r/n). The partitions \underline{ν} with ν ∈ Γ∩W_{p}·0 may be recursively defined as follows: Suppose this notation is understood for all small partitions with at most n parts. Then ν ∈ Γ ∩ W_{p} · 0 if and only if there is a partition \underline{ω} ◁ \underline{ν}, with ω ∈ Γ ∩ W_{p} · 0, such that, for some 0 ≤ i < j ≤ n and 0 < m, we have:

(a) \underline{ν}_{i} = \underline{ω}_{i} + m and \underline{ν}_{j} = \underline{ω}_{j} − m;

10The argument given in [CPS5] is incomplete when n = p, unless n = r. However, because of the assumption that p > 3, the argument is easily repaired using [HN, (3.3.4)] or the i = 1 case of [PS, Thm. 4.1]. In fact, these more recent results imply that the assumption that p ≥ n may even be dropped in the equivalence (5.0.3), when p > 3. A related version of (5.0.3) for p > n is essentially a result in [E1, p. 124], restated in [CPS5, ftn. 15].

(b) \underline{ν}_{i} − \underline{ν}_{j} + j − i ≡ m mod p. The above conditions are just a translation of conditions for ω to be related to ν by a certain kind of reflection. Notice that \underline{ν}_{i} − \underline{ν}_{j} + j − i < np < \underline{ω}_{i} − \underline{ω}_{j} + j − i for some integer n. If the pair i, j are unique with this property, and only m < p works in (a) above, then ω = νs (and conversely, if \underline{ω} > \underline{ν}).

Given \underline{μ} ∈ Λ^{+}(n, r), μ ∈ Γ_{Jan} if and only if

\underline{μ}_{1} − \underline{μ}_{n} + n − 1 ≤ p(p − n + 2). (5.1.2)

As before, we will consider if the Lusztig character formula holds for Γ ∩ W_{p} · 0 for any ideal Γ in X^{+}.

There is a natural and unique submodule T_{\underline{μ},<\underline{γ}}, for any \underline{γ} ∈ Λ^{+}(r), which has a filtration by Specht modules S\underline{τ} with \underline{τ} ◁ \underline{γ}, and with T\underline{μ}/T_{\underline{μ},<\underline{γ}} filtered by S\underline{τ} with \underline{τ} not smaller that \underline{γ}. The existence and uniqueness of T_{\underline{μ},<\underline{γ}} follow from (5.0.3) and the existence and uniqueness of the module P_{\underline{μ},<\underline{γ}} from §4 (which follows from well-known quasi-hereditary algebra theory).^{11}

Theorem 5.2. Assume that p ≥ n and let Γ be a finite ideal in X^{+}. Then the Lusztig character formula holds for Γ ∩ W_{p} · 0 if and only if for each map S\underline{λ} → T_{\underline{μ},<\underline{λs}}, which is not a split injection, the induced map

Ext_{k𝔖_{r}}^{1}(S\underline{λs}, S\underline{λ}) → Ext_{k𝔖_{r}}^{1}(S\underline{λs}, T_{\underline{μ},<\underline{λs}}) (5.2.1)

on Ext^{1}-groups is zero, whenever μ < λ < λs in Γ ∩ W_{p} · 0 (for some s ∈ S_{p}).

Proof. This follows from Thm. 4.2 and (5.0.2) using the contravariant equivalence of exact categories (5.0.3). □

Using Lemma 5.1, we can restate the above result in a slightly different, but suggestive way.

Theorem 5.3. Assume that p ≥ n and let Γ be a finite ideal in X^{+}. Then the Lusztig character formula holds for Γ ∩ W_{p} · 0 if and only if, whenever μ < λ < λs belong to Γ ∩ W_{p} · 0 (for some s ∈ S_{p}), any morphism S\underline{λ} → T_{\underline{μ},<\underline{λs}} which is not a split injection, fits into a commutative diagram

\begin{aligned}0 & \overline{} & S\underline{λ}_{} & [figure] & E & \overline{} & S\underline{λs} & \overline{} & 0 \\ & & T_{\underline{μ},<\underline{λs}} \end{aligned} (5.3.1)

11There is always a natural construction of T_{\underline{μ},<\underline{γ}}, obtained by reduction mod p from a similar module defined over a principal ideal domain Z of characteristic zero. See [CPS5, Thm. 5.2.1] for the existence of the relevant filtrations over Z, which uses no algebraic group theory. The uniqueness of the analogue of T_{\underline{μ},<\underline{γ}} over Z is easy. Indeed, over Z, T_{\underline{μ},<\underline{γ}} may be described as the intersection of the (Z-version) of T\underline{μ} with an evident canonical module defined over the quotient field of Z.

of k𝔖_{r}-modules, in which the horizonal row is the unique (up to scalar equivalence) short exact sequence defined in (5.1.1).

Taking Γ = Γ_{Jan}, the above theorems present a specific necessary and sufficient condition in terms of symmetric group modules for the validity of the Lusztig conjecture for SL_{n}(k). This result should be compared to [PS, Prop. 7.1], where the authors proved a sufficient condition for the validity of the Lusztig conjecture for SL_{n}(k). The condition involved only the cohomology of symmetric groups, stating that the Lusztig conjecture was true provided certain groups Ext_{k𝔊_{s}}^{1}(S\underline{τ}, D\underline{σ}) were equal to 0. Here the (usually large) integer s is defined in terms of n and the weights Γ_{Jan} ∩W_{p} ·0. Also, \underline{τ}, \underline{σ} are various (regular) partitions of s indexed by the set Γ_{Jan} ∩ W_{p} · 0, which are defined using the Erdmann function d in [PS, (6.1.2)]. We also have the following result which is in the spirit of [PS].

Proposition 5.4. Assume that p ≥ n. Let Γ ⊂ X^{+} be a finite ideal as before. Suppose, for fixed λ < λs as above, that (5.2.1) fails for some μ < λ. Let μ < λ be maximal so that (5.2.1) fails. Assume that μ, λ and λs are all p-restricted. Then Ext_{k𝔖_{r}}^{1}(S\underline{μ}′, D_{(\underline{λs})′}) ≠ 0 and Ext_{k𝔖_{r}}^{1}(S\underline{μ}′, D_{\underline{λ}}′) ≠ 0.

Proof. By [DEN, Thm. (4.3)] (or [PS, Cor. 5.3]),

(5.4.1) dim Ext_{S(r,r)}^{1}(Δ(\underline{λ}), L(\underline{μ})) ≤ dim Ext_{k𝔖_{r}}^{1}(S_{\underline{λ}}′, D\underline{μ}′),

whenever the partitions \underline{λ}′ and \underline{μ}′ are p-regular. For any partition \underline{λ}, the weight λ ∈ X^{+} is restricted if and only if the dual partition \underline{λ}′ is p-regular. Now the result follows from (5.4.1), together with Thm. 4.2 and (5.0.2). □

Remarks 5.5. (a) Assume that Γ = Γ_{Jan} and that p ≥ 3p − 2 so that all restricted weights lie in Γ_{Jan}. By [K] (see §3), to see that the Lusztig character formula holds for Γ_{Jan} ∩ W_{p} · 0, it is enough to check it for restricted regular weights. Then the condition that λ, λs be restricted in the statement of Prop. 5.4 is not so severe: Let τ ∈ X^{+} ∩ W_{p} · 0, and suppose τ = λs for some λ ∈ X^{+}, λ < τ. Further, if τ is restricted, then λ is also restricted. This follows since λ and τ are separated by only one affine hyperplane H_{α,mp}, and H_{α,mp} separates τ from 0. So H_{α,np} is not among those hyperplanes defining the restricted region. If the Lusztig character formula holds for each weight in X^{+} ∩ W_{p}.0 smaller than τ, then Thm. 4.1(b) shows that it holds for τ if and only the inequality (4.1.1) holds. Thus, in Prop. 5.4, we can assume to start that both λ and λs restricted. (However, it still may be true that μ is not restricted. The Erdmann function d [PS, (6.1.2)] in offers some way around this problem, but it would seem best to first check if the conclusion of Prop. 5.4 holds when all three weights involved are restricted.)

(b) The results of this section avoid the use of the function d altogether. Thus, they give a way of using a smaller symmetric group than in Erdmann’s paper [E2] to determine, say, the decomposition numbers for the “principal” block (containing the determinant module) of the Schur algebra S(n, r) in terms of 𝔖_{r}, without increasing r.

However, it has been long known that the decomposition numbers of the endomorphism ring of a permutation module can be equated with multiplicities of ordinary characters inside those of p-adic indecomposable components of the permutation module. See [S1]. So, Schur algebra decomposition numbers can be computed in terms of symmetric group permutation module theory in this way, though this method would require knowledge the indecomposable components of permutation modules. Erdmann’s reduction notably uses (albeit after a big increase in r) just the projective indecomposable components, whose multiplicities are actual symmetric group decomposition numbers. Perhaps the notable aspect of our reduction, aside from the fact that no increase in r is required, is that no indecomposable components of permutation modules are required at all, only certain submodules of permutation modules whose definition is transparent in terms of ordinary character theory.

PART III: Levi subgroups and irreducible modules

Again, G is a simple, simply connected algebraic group over k. When p ≥ h, the discussion will often make use of the notion, defined in §3, of what it means for the Lusztig character formula to hold for a subset Σ ⊆ W_{p} · 0.

6. A theorem of Hemmer and a generalization

It is natural to ask for a version of (1.1.4) relating the groups Ext_{G}^{•}(L(λ), L(μ)) to the groups Ext_{H}^{•}(L_{H}(λ), L_{H}(μ)). This section considers this question. Thm. 6.1 below establishes an inequality for Ext^{1}-groups which is in the spirit of (1.1.4) and (1.1.5). A version of this result for quantum enveloping algebras at a root of unity is easily obtained along the same lines.

In [H, Thm. 2.3], Hemmer proves that if λ, μ ∈ Λ^{+}(n, r) satisfy λ_{1} = μ_{1} = m then there is an injection

Ext_{S(n,r)}^{1}(L(λ), L(μ)) ↪ Ext_{S(n−1,r−m)}^{1}(L(λ̄)), L(μ̄)), (6.0.1)

where λ̄ is the partition of r − m obtained by removing λ_{1} from λ, and μ̄ is defined similarly. This result is then interpreted in terms of comparing the Ext-quiver of S(n, r) with that of S(n − 1, r − m).

We can generalize Hemmer’s theorem to other types:

Let H be a Levi subgroup of G and Ω = ω + ℤΦ_{H} is a coset of ℤΦ_{H} in ℤΦ. Put Ω^{+} = Ω ∩ X^{+}.

Theorem 6.1. Let λ, μ ∈ Ω^{+}. Then there is a natural injection

Ext_{G}^{1}(L(λ), L(μ)) ↪ Ext_{H}^{1}(L_{H}(λ), L_{H}(μ)). (6.1.1)

Proof. We will argue that the required injection is induced by the truncation functor π_{Ω} (1.0.1). We can assume that λ ≤ μ. There is a natural short exact sequence 0 → Q(λ) → Δ(λ) → L(λ) → 0. The long exact sequence of Ext_{G}^{•} with respect to the functor Hom_{G}(−, L(μ)) gives an exact sequence

Hom_{G}(Q(λ), L(μ)) → Ext_{G}^{1}(L(λ), L(μ)) → Ext_{G}^{1}(Δ(λ), L(μ)).

All the composition factors L(τ) of Q(λ) satisfy τ < λ < μ; so Hom_{G}(Q(λ), L(μ)) = 0. A similar conclusion holds for the analogous exact sequence 0 → Q_{H}(λ) → Δ_{H}(λ) → L_{H}(λ) → 0. Therefore, the functor π_{Ω} provides a commutative diagram

0 → Ext_{G}^{1}(L(λ), L(μ)) → Ext_{G}^{1}(Δ(λ), L(μ))

[figure: π_{Ω}; π_{Ω}]
0 → Ext_{H}^{1}(L_{H}(λ), L_{H}(μ)) → Ext_{H}^{1}(Δ_{H}(λ), L_{H}(μ))

with exact rows. By (1.1.2) for M = L(μ) (or [CPS6, Cor. 10]), the right hand vertical map is an isomorphism. Hence, the left hand vertical map is an injection, as required. □

7. An example

In general, the inclusion map Ext_{G}^{1}(L(λ), L(μ)) ↪ Ext_{H}^{1}(L_{H}(λ), L_{H}(μ)) given in Thm. 6.1 need not be an isomorphism.^{12}

Let G = SL_{3}(k), with char k = p = 3. We list the dominant weights as in [Bo, p. 250]. The Lusztig conjecture is known to be true for G. In particular, by the result of Kato discussed in §3, the Lusztig character formula holds for 4ϖ_{1} + ϖ_{2} = (ϖ_{1}+ϖ_{2})+3ϖ_{1}, since ϖ_{1}+ϖ_{2} is a restricted weight in W_{p}·0 and ϖ_{1} ∈ \overline{C^{+}}. Similarly, it holds for 3ϖ_{1}. Thus, the Lusztig character formula holds for each weight in the sequence 0 < ϖ_{1}ϖ_{2} < 3ϖ_{1} < 4ϖ_{1} + ϖ_{2} of adjacent dominant weights. On the other hand, Lemma 3.1 (or an unpleasant computation) shows that the Lusztig character formula fails for 3ϖ_{1} + 3ϖ_{2} = (4ϖ_{1} + ϖ_{2})s_{α_{2}}. By Thm. 4.1(c), Ext_{G}^{1}(L(4ϖ_{1} + ϖ_{2}), L(3ϖ_{1} + 3ϖ_{2})) = 0. However, let H be the Levi subgroup with Φ_{H} = {α_{2}} (i. e., defined by Π_{1} in the notation of §2), then, directly or using Thm. 4.1(c) again, we have that Ext_{H}^{1}(L_{H}(4ϖ_{1} + ϖ_{2}), L_{H}(3ϖ_{1} + 3ϖ_{2})) ≠ 0.

In terms of the group GL_{3}(k), let λ = (6, 2, 1), μ = (6, 3, 0) ∈ Λ^{+}(3, 9). In the notation of §2, Example 1,

Ext_{GL_{3}(k)}^{1}(L(λ), L(μ) = 0,

12The examples in this section provide a (negative) answer to a question raised by Hemmer in his lecture at the AMS Conference “Representations of algebraic groups, quantum groups, and Lie algebras,” Snowbird, Utah, July 2004. The form of the examples suggest a positive answer to Hemmer’s question is almost equivalent to the validity of the Lusztig conjecture.

while

Ext_{H}^{1}(L(\overline{λ}), L_{H}(\overline{μ})) ≠ 0.

Therefore, the map (6.0.1) is not an isomorphism in this case.

Another, similar but interesting example, is provided by GL_{5}(k) when p = 5, taking λ = (10, 5^{3}), (10, 5^{2}, 4, 1) ∈ Λ^{+}(5, 25). (From computer results discussed in [S2], and the translation principle, (3.0.2) holds in the category C_{G} for any regular restricted weight ν.) In fact, this last example let us to the “smaller” one above.

Despite these examples, §8 below shows that (6.1.1) is an isomorphism for regular weights lying in a finite ideal Γ with the property that the Lusztig character formula holds for Γ ∩ W_{p} · 0.

8. An Ext^{1}-isomorphism result and higher Ext^{•}-groups

In this section, we take up, for arbitrary cohomological degree n, the study of the map Ext_{G}^{n}(L(λ), L(μ)) → Ext_{H}^{n}(L_{H}(λ), L_{H}(μ)) induced by the truncation functor π_{Ω}. We will show that if λ, μ are regular weights, lying in a finite ideal Γ such that the Lusztig character formula holds for Γ_{reg}, then π_{Ω} always induces an surjection provided λ, μ belong to the same ℤΦ_{H}-coset. In particular, (6.1.1) is an isomorphism in this case. To obtain such a result, it will be necessary to work in the context of the homological dual of a quasi-hereditary algebra, and so we begin by recalling some general results.

In [CPS3], the homological dual A^{!} of a quasi-hereditary algebra A was defined to be the Yoneda Ext-algebra

A^{!} = Ext_{A}^{•}(L_{0}, L_{0})

where L_{0} is the direct sum of the distinct irreducible A-modules. If C = A–mod, write C^{!} for A^{!}–mod.

The rest of this section will make use of properties of a highest weight category C ≅ A–mod having a Kazhdan-Lusztig theory with respect to a length function l : Λ → ℤ^{+} on its weight poset Λ. This means that, for any λ, μ ∈ Λ,

(8.0.2)  \begin{cases}Ext_{C}^{n}(L(λ), ∇(μ)) ≠ 0 \\ or \\ Ext_{C}^{n}(Δ(μ), L(λ)) ≠ 0 \end{cases} ⟹ n ≡ l(λ) − l(μ) mod 2

See [CPS2, §3] or [CPS3, §1.3] for further discussion. We will always assume below that C has finite poset Λ, so that C ≅ A–mod for some finite dimensional algebra A. The following result, proved in [CPS3], summarizes some basic properties of the homological dual.

Theorem 8.1. Assume that C ≅ A–mod has a Kazhdan-Lusztig theory with respect to a length function l : Λ → ℤ^{+}. Let Ξ be a coideal in Λ. Then:

(a) A^{!} is a quasi-hereditary algebra; more precisely, C^{!} = A^{!}–mod is a highest weight category with weight poset Λ^{op} (the poset opposite to Λ).

(b) C(Ξ) has a Kazhdan-Lusztig theory with respect to l|_{Ξ}.

(c) C(Ξ)^{!} ≅ C^{!}[Ξ^{op}] as highest weight categories.

Proof. (a) is proved in [CPS3, Thm. 2.1]. (b) is well-known, but follows directly in the spirit of this paper as follows: Let j^{∗} : C → C(Ξ) be the natural quotient functor. By [CPS6, Lemma 6], given λ, μ ∈ Ξ, Ext_{C(Ξ)}^{•}(L_{Ξ}(λ), ∇_{Ξ}(μ)) ≅ Ext_{C}^{n}(L(λ), ∇(μ)) and Ext_{C(Ξ)}^{•}(Δ_{Ξ}(μ), L_{Ξ}(λ)) ≅ Ext_{C}^{•}(Δ(μ), L(λ)), where we have written L_{Ξ}(λ), etc. for the irreducible, etc. object in C(Ξ) indexed by λ. Now (b) follows from the definition (8.0.2). For (c), see [CPS3, (2.3)]. □

In (c) above, C(Ξ)^{!} ≅ (A_{Ξ})^{!}–mod, where A_{Ξ} is a quasi-hereditary algebra such that A_{Ξ}–mod ≅ C(Ξ). The proof given in [CPS3, p. 308] is more precise, showing that the natural algebra homomorphism

(8.1.1) Φ : A^{!} = Ext_{C}^{•}(L_{0}, L_{0}) → Ext_{C(Ξ)}^{•}(j^{∗}L_{0}, j^{∗}L_{0}) = (A_{Ξ})^{!},

induced by the quotient functor j^{∗} : C → C(Ξ), defines an equivalence

Φ^{∗} : (A_{Ξ})^{!}–mod → A^{!}/J–mod

of highest weight categories, for some (idempotent) ideal J (denoted A^{!}s_{Γ}A^{!} in [CPS3]) which is part of a defining sequence for A^{!}. Thus, A^{!}/J ≅ (A_{Ξ})^{!}. In particular, we have:

Corollary 8.2. Assume that A–mod has a Kazhdan-Lusztig theory. Then the map Φ defined in (8.1.1) is surjective.

We now return to the setting of a simple, simply connected algebraic group G over k, assuming that p ≥ h. Fix a finite ideal Γ ⊆ X^{+}, and let C_{G,0}[Γ] denote the highest weight category consisting of all finite dimensional G-modules with composition factors L(ξ) with ξ ∈ Γ∩W_{p} ·0.^{13} Define a length function l : Γ∩W_{p} ·0 → ℕ by l(w ·0) = l(w). By the argument in [CPS2, Thm. 5.3] (see the discussion of Thm. 4.1), C_{G,0}[Γ] has a Kazhdan-Lusztig theory if and only if the Lusztig character formula holds for Γ∩W_{p}·0.

Let H be a Levi subgroup of G as in §1. In Thm. 1.1, Γ_{F}^{+} can be replaced by a “block analog”. That is, let B be any union of W_{p}-orbits in the dot action on X. Put Γ_{F,B}^{+} = Γ_{F}^{+} ∩ B. Then clearly Ω_{F,B}^{+} := Γ_{F,B}^{+} ∩ Ω is a union of dominant weights in orbits of the affine Weyl group W_{H,p} of H. (Use the same weight ρ in defining the dot action of W_{H,p} as is used for W_{p}.) Thm. 1.1 remains true with Γ_{F}^{+} and Ω_{F}^{+} replaced by Γ_{F,B}^{+} and Ω_{F,B}^{+}, respectively.

We will take B = X_{reg}^{+}, the set of regular dominant weights, writing Γ_{F,reg}^{+} and Ω_{F,reg}^{+} for Γ_{F,B}^{+} and Ω_{F,B}^{+}, respectively,

Fix a finite ideal Σ in X^{+} such that the Lusztig character formula holds for Σ_{reg} ≠ ∅. Let Ω = ω +ℤΦ_{H} be a coset of ℤΦ_{H} in X, with ω ∈ Σ. Put F = Σ∩Ω and, as in Thm. 1.1, form Γ_{F}^{+} and Ω_{F}^{+}. Thus, Ω_{F}^{+} is a coideal in Γ_{F}^{+} and an ideal in (X^{(H)+}, ≤). Set Λ_{0} = Γ_{F,reg}^{+} and Ω_{0} = Ω_{F,reg}^{+}, and form the highest weight categories C_{G,0} = C_{G}[Λ_{0}] and C_{H,0} = C_{H}[Ω_{0}]. Let L_{0} be the direct sum of the distinct irreducible G-modules L(λ) for λ ∈ Λ_{0}. Similarly, let L_{0,H} be the direct sum of the distinct irreducible H-modules L_{H}(λ) for λ ∈ Ω_{0}.

13If we replace the ≤ partial ordering on X^{+} by the Jantzen ↑ ordering, then Γ ∩ W_{p} · 0 becomes an ideal in (X^{+}, ↑).

Theorem 8.3. Assume the notation of the previous paragraph. Then:

(a) The category C_{H,0} has a Kazhdan-Lusztig theory. In particular, the algebra A_{H}^{!} := Ext_{H}^{•}(L_{0,H}, L_{0,H}) is quasi-hereditary.

(b) The natural map

A^{!} = Ext_{G}^{•}(L_{0}, L_{0}) → Ext_{H}^{•}(L_{0,H}, L_{0,H}) = A_{H}^{!}

induced by π_{Ω} is surjective, and its kernel is a defining ideal J of the quasi-hereditary algebra A^{!}.

(c) We have A^{!}/J–mod ≅ C_{G,0}^{!}[Ω_{0}^{op}].

Proof. (a) follows from Thm. 8.1, using Thm. 1.1 which shows that C_{H,0} ≅ C_{G,0}(Ω_{0}). Then (b) and (c) follow from Cor. 8.2. □

Remarks 8.4. (a)A somewhat less precise way to state part (c) is, in view of part (b),

C_{G,0}^{!}[Ω_{0}^{op}] ≅ C_{H}[Ω_{0}]^{!}.

(b) As remarked earlier, for semisimple algebraic groups having a fixed root system, the Lusztig conjecture is known to hold provided that the characteristic p is large enough, though lower bound is known on the size of p in general. Thus, in those cases, we can in Thm. 8.3 take Σ = Γ_{Jan}, the Jantzen region.

Corollary 8.5. Assume the set-up in the paragraph immediately before Thm. 8.3. Let λ, μ ∈ Σ_{reg} ∩ ω + ℤΦ_{H}. Then

(a) Ext_{G}^{1}(L(λ), L(μ)) ≅ Ext_{H}^{1}(L_{H}(λ), L_{H}(μ)).

(b) The map Ext_{G}^{n}(L(λ), L(μ)) → Ext_{H}^{n}(L_{H}(λ), L_{H}(μ)) which is induced by π_{Ω} is surjective for all n ≥ 0. In particular,

dim Ext_{G}^{n}(L(λ), L(μ)) ≥ dim Ext_{H}^{n}(L_{H}(λ), L_{H}(μ)).

Proof. Part (b) follows directly from Thm. 8.3. Part (a) is then a consequence of (b) for n = 1 and Thm. 6.1. □

Remarks 8.6. (a) An alternative proof of the inequality in Cor. 8.5(b) can be based on the Ext-formula proved in [CPS2, Thm. 3.5], which states that

dim Ext_{C}^{n}(L(λ), L(μ)) = ∑_{m=0}^{n} ∑_{ν} dim Ext_{C}^{m}(L(λ), ∇(ν)) · dim Ext_{C}^{n−m}(Δ(ν), L(μ))

in any highest weight category C with a Kazhdan-Lusztig theory. This formula shows just what part of dim Ext_{G}^{n}(L(λ), L(μ)) contributes of Ext_{H}^{n}(L_{H}(λ), L_{H}(μ))—namely, the weight ν must lie in Ω.

(b) Now we consider Hemmer’s injective map (6.0.1) in the case when the Lusztig conjecture is true for SL_{n}. Assume that λ, μ ∈ Λ^{+}(n, r) satisfy λ_{1} = μ_{1} = m, and let λ̄, μ̄ be as before. Assume that λ_{i}−λ_{j} ≢ i−j and μ_{i}−μ_{j} ≢ i−j mod p for all i < j ≤ n (see (5.0.1)) and that λ_{1} − λ_{n} + n − 1 ≤ p(p − n + 2) and μ_{1} − μ_{n} + n − 1 ≤ p(p − n + 2) (see (5.1.2)). Then the map (6.0.1) is an isomorphism. Moreover, for higher degree Ext-groups, there is a corresponding surjection, a somewhat surprising turnabout.

(c) Observe that an alternative proof of Cor. 8.5(a) can be based on the commutative diagram in the proof of Thm. 6.1, once it is observed that [CPS2, Thm. 4.3] implies that the horizonal map is surjective.

Corollary 8.7. Let Σ_{reg} be as in the set-up in the paragraph immediately before Thm. 8.3, and put X := Σ_{reg}. (In particular, the Lusztig character formula holds for irreducible modules L(λ), λ ∈ X.) Let C_{H}[X] denote the category of all finite dimensional rational H-modules which have composition factors L_{H}(ξ), ξ ∈ X. Then C_{H}[X] is a highest weight category with respect to the poset (X, ≤_{H}), and, further, C_{H}[X] has a Kazhdan-Lusztig theory in the sense of (8.0.2).

Proof. First note, that if Δ_{H}(ν) is a standard module for ν ∈ X, and L_{H}(ϖ) is a composition factor, then ϖ ∈ X^{+} since X^{+} is an ideal in (X^{(H)}, +, ≤_{H}). Also, L(ω) is a composition factor of Δ(ν)—this is a result of Donkin [D1]—see also [CPS6, Cor. 12]. In particular, ν and ϖ belong to the same block of G, and so ν and ϖ are conjugate under the dot action of W_{p}. Hence, ϖ ∈ X. Now consider the projective cover P_{Γ,H}(ϖ) of L_{H}(ϖ) in the category C_{H}[Γ], where Γ is the set of dominant weights (for G) regarded as an ideal in (X^{(H)+}, ≤_{H}). If Δ_{H}(ξ) is a Δ_{H}-section, then L_{H}(ϖ) is a composition factor of ∇_{H}(ξ) by Brauer-Humphreys reciprocity [CPS1, Thm. 3.11]. Hence, L_{H}(ϖ) is a composition factor of Δ_{H}(ξ). As before, this forces ξ ∈ X. Thus, C_{H}[X] is a highest weight category.

The final assertion follows from Thm. 8.3(a), since Ext_{H}^{•} vanishes between modules with respective composition factors in different cosets of the root lattice of H. □

Finally, we have the following general (and elementary) result which shows that the Ext^{1}-isomorphism property behaves well with respect to Frobenius twisting.

Theorem 8.8. Assume that if G or H has a component of type C_{m}, then p > 2. Suppose that λ, μ lie in the same ℤΦ-coset Ω = ω + ℤΦ_{H} and that π_{Ω} induces an isomorphism Ext_{G}^{1}(L(λ), L(μ)) →^{∼} Ext_{H}^{1}(L_{H}(λ), L_{H}(μ)). For any positive integer r, put Ω^{(r)} = p^{r}ω + ℤΦ_{H}. Then π_{Ω^{(r)}} induces an isomorphism Ext_{G}^{1}(L(p^{r}λ), L(p^{r}μ)) →^{∼} Ext_{H}^{1}(L_{H}(p^{r}λ), L_{H}(p^{r}μ)).

Proof. It suffices to consider the r = 1 case. The hypothesis on p guarantees that, if G_{1} denotes the first Frobenius kernel of G, then H^{1}(G_{1}, k) = 0 [Jan, Prop. 12.9]. Hence, by a Hochschild-Serre spectral sequence argument for G_{1} ⊲ G, Ext_{G}^{1}(L(pλ), L(pμ)) ≅ Ext_{G}^{1}(L(λ), L(μ)). Similar remarks apply to Ext_{H}^{1}(L_{H}(pλ), L_{H}(pμ)). Therefore,

dim Ext_{G}^{1}(L(pλ), L(pμ)) = dim Ext_{H}^{1}(L_{H}(pλ), L_{H}(pμ))

and the result follows from Thm. 6.1, since pλ, pμ ∈ Ω^{(1)}. □

Thus, in general, the injection (6.1.1) is an isomorphism for some regular weights well outside Γ_{Jan}, where the Lusztig character formula fails.^{14}

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

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