ON Ext-TRANSFER FOR ALGEBRAIC GROUPS

EDWARD T. CLINE, BRIAN J. PARSHALL, AND LEONARD L. SCOTT

We dedicate this paper to the memory of our friend, Gordon Keller

Abstract. This paper builds upon the work of Cline [C] and Donkin [D1] to describe explicit equivalences between some categories associated to the category of rational modules for a reductive group G and categories associated to the category of rational modules for a Levi subgroup H. As an application, we establish an Ext-transfer result from rational G-modules to rational H-modules. In case G = GL_{n}, these results can be illustrated in terms of classical Schur algebras. In that case, we establish another category equivalence, this time between the module category for a Schur algebra and the module category for a union of blocks for a natural quotient of a larger Schur algebra. This category equivalence provides a further Ext-transfer theorem from the original Schur algebra to the larger Schur algebra. This result, announced in [PS2, (6.4b)], extends to the category level the decomposition number method of Erdmann [E2]. Finally, we indicate (largely without proof) some natural variations to situations involving quantum groups and q-Schur algebras.

Research supported in part by the National Science Foundation.

1. Introduction

Let G be a reductive algebraic group over an algebraically closed field k. Let H be a Levi subgroup of G. In independent work, Cline [C] and Donkin [D1] present different ways to relate the representation theory of H to that of G. Actually, the approach in [C] works with infinitesimal thickenings G_{r}T and H_{r}T (r ≥ 1) (defined by a Frobenius map F : G → G) and uses Harish-Chandra induction to pass up from H_{r}T to G_{r}T. The approach in [D1] proceeds in the opposite direction, coming down to H from G by a weight truncation operator (first studied by Jantzen [Jan1], [Jan2]). In both approaches, a fixed coset Ω := ν + ℤΦ_{H} of the root lattice of H in the weight lattice X of G plays a key role. In [C], there is a full embedding from the derived category of rational H_{r}T -modules whose composition factors have high weights in Ω into the derived category of rational G_{r}T -modules. Thus, under this embedding Ext^{•} groups are preserved. The theory in [D1] is more character-theoretic, using the natural truncation operator π_{Ω} on weights of G-modules, throwing away weights not in Ω: this operation preserves irreducibility, standard modules, and decomposition numbers.

The main goal of this paper is to establish that the above truncation procedure also has some remarkable homological properties. These properties, in turn, derive from an equivalence between a certain category of rational G-modules and a category of rational H-modules. The homological properties, of considerable interest in their own right, also contain Donkin’s result on decomposition numbers, which may now be seen in terms of an Euler characteristic property.

Our paper is organized as follows. Section 2 collects together some preliminary material concerning general highest weight categories. In Section 3, we prove the main results just mentioned. Section 4 shows that a special case of the procedure amounts to passage from homogeneous polynomial modules of degree r for GL_{r}(k) to homogeneous modules of degree r for GL_{n}(k), i. e., passage from S(r, r)–mod to S(n, r)–mod, when n ≤ r. Here S(n, r) denote the Schur algebra of bidegree (n, r) over the field k. In this way, the results of Section 3 cast the classical Schur algebra situation in a broader context. Some of the decomposition number applications in [D1] foreshadow this context, though our interests here are more homological.

Section 5 presents a different kind of Ext-transfer result that arises in the case of Schur algebras, i. e., when G ≅ GL_{n}(k). This transfer, which was discussed in [PS2, §6] in special cases, actually arises from an interesting (and new) Morita equivalence between the Schur algebra S(r, r) and an algebra which is a product of various blocks of a natural quotient algebra of S(s, s), where s = pr + (p − 1)\frac{r(r−1)}{2}; see Thm. 17. Then Cor. 18 produces, by means of a standard recollement argument, a further Exttransfer. As suggested already for the more special results in [PS2], this isomorphism of Ext-groups provides a natural cohomological analogue of Erdmann’s decomposition number method [E2]. The present paper even extends this to the categorical level.

Finally, we briefly indicate in Section 6 how some of these ideas extend to the context of quantum groups and q-Schur algebras.

We thank both referees of this paper for providing detailed comments which the authors found useful.

2. Preliminaries

1. Quasi-hereditary algebras. We assume familiarity with the basic theory of quasihereditary algebras and highest weight categories [CPS1]. Actually, it is convenient and simplifying in our context to consider finite dimensional modules only. Thus, “highest weight category” means the category of finite length objects in a highest weight category in the sense of [CPS1]. This convention will be convenient, for example, when dealing with the category of rational modules for a reductive algebraic group.

Let A be a quasi-hereditary algebra over an algebraically closed field k. Thus, the category of finite dimensional A-modules, denoted A–mod, is a highest weight category with respect to a poset (Λ, ≤) (i. e., the weight poset). We remark that in this case, A–mod is a highest weight category in the full sense of [CPS1].

The set Λ indexes the set of isomorphism classes of irreducible objects in A-mod. For λ ∈ Λ, let L(λ) = L_{A}(λ) be an irreducible A-module representing the class indexed by λ and Δ(λ) = Δ_{A}(λ) (resp., ∇(λ) = ∇_{A}(λ)) denote the corresponding standard (resp., costandard) objects in A-mod. Thus Δ(λ) has head L(λ) and ∇(λ) has socle L(λ).

Let C be an arbitrary highest weight category with weight poset Λ. Recall that two indecomposable objects M, M′ in C belong to the same block provided there is a finite sequence M = M_{1}, M_{2}, …, M_{n} = M′ of indecomposable objects in C so that each adjacent pair M_{i}, M_{i+1} shares a common composition factor. At the level of the weight poset, a subset B ⊆ Λ is called a block provided the set of irreducible objects L(ξ) with ξ ∈ B comprise the set of irreducible modules in a block of C.

The following result will be useful; see [CPS3, Lemma 2.2].

Lemma 1. Let A be a quasi-hereditary algebra. For λ, μ ∈ Λ,

dim Ext_{A}^{i}(Δ(λ), ∇(μ)) = δ_{λ,μ}δ_{i,0}.

In establishing our main results, we will require the following Comparison Theorem [PS1, Thm. 5.8] stated below. Let A_{1}, A_{2} be quasi-hereditary algebras over an algebraically closed field k. Suppose that sets of isomorphism classes of irreducible objects in A_{1}–mod and A_{2}–mod are both indexed by the same set Λ, and that ≤ is a poset structure on Λ making both A_{1}–mod and A_{2}–mod into highest weight categories. For i = 1, 2, and λ ∈ Λ, let L_{i}(λ) = L_{A_{i}}(λ), Δ_{i}(λ) = Δ_{A_{i}}(λ) and ∇_{i}(λ) = ∇_{A_{i}}(λ) respectively.

Theorem 2. Let E : A_{1}−mod → A_{2}−mod be an exact, additive functor. Suppose that for all λ ∈ Λ,

E∇_{1}(λ) ≅ ∇_{2}(λ) and EΔ_{1}(λ) ≅ Δ_{2}(λ).

Then E is an equivalence of categories. In particular, the algebras A_{1} and A_{2} are Morita equivalent.

2. Some homological algebra. Let A be a quasi-hereditary algebra over an algebraically closed field k. Let Γ be an ideal in the weight poset Λ (i. e., μ ≤ λ ∈ Γ ⇒ μ ∈ Γ), and denote the coideal Λ \ Γ by Ω. Let A–mod[Γ] be the full subcategory of A–mod having as objects those A-modules M all of whose composition factors are isomorphic to various L(λ), λ ∈ Γ. Let i_{∗} : A–mod[Γ] → A–mod be the natural exact full embedding. Then A–mod[Γ] is a highest weight category with poset Γ; in addition, A–mod[Γ] ≅ A/J–mod for some idempotent ideal J ⊴ A. Also, A–mod[Γ] is a Serre subcategory of A–mod; the quotient category of A–mod by A–mod[Γ] is denoted by A–mod(Ω) because it is a highest weight category with weight poset Ω. Let j^{∗} : A–mod → A–mod(Ω) be the canonical exact quotient functor. If J = AeA for an idempotent e ∈ J, then A–mod(Ω) ≅ eAe–mod. Under this identification, the functor j^{∗} identifies with the functor A–mod → eAe–mod which sends an A-module M to the eAe-module eM. (See [P, §§1,2] for further discussion.) In addition, j^{∗}M = 0 if and only if M lies in the image of the functor i_{∗}.

For γ ∈ Γ (resp., ω ∈ Ω), we write L_{Γ}(γ), Δ_{Γ}(γ), and ∇_{Γ}(γ) (resp., L_{Ω}(ω), Δ_{Ω}(ω) and ∇_{Ω}(ω)) for the corresponding irreducible, standard and co-standard objects in A/J–mod ≅ A–mod[Γ] (resp., A–mod(Ω) ≅ eAe−mod). For γ ∈ Γ, we have i_{∗}L_{Γ}(γ) ≅ L(γ), i_{∗}Δ_{Γ}(γ) ≅ Δ(γ) and i_{∗}∇_{Γ}(γ) ≅ ∇(γ). Also, for ω ∈ Ω, j^{∗}L(ω) ≅ L_{Ω}(ω), j^{∗}Δ(ω) ≅ Δ_{Ω}(ω) and j^{∗}∇(ω) ≅ ∇_{Ω}(ω).

This set-up leads to a “recollement diagram”:

(1) D^{b}(A–mod[Γ]) \underleftarrow{i^{∗}} ; \underrightarrow{i_{∗}} ; \underleftarrow{i^{!}} D^{b}(A–mod) \underleftarrow{j_{!}} ; \underrightarrow{j^{∗}} ; \underleftarrow{j_{∗}} D^{b}(A–mod(Ω))

for the bounded derived categories of the respective derived categories. Here i_{∗} and j^{∗} denote the functors on derived categories induced by the exact functors i_{∗} and j^{∗} on the module categories. In particular, if X^{•} belongs to D^{b}(A/J−mod), the i_{∗}X^{•} is obtained by applying i_{∗} to each term of the complex X^{•}. Similarly, given a complex Y^{•} ∈ D^{b}(A–mod), j^{∗}Y^{•} ∈ D^{b}(eAe–mod) = D^{b}(A–mod(Ω)) is represented by the complex eX^{•}. The functor i_{∗} is a full embedding of derived categories; thus,^{1}

(2) Ext_{A/J}^{•}(M, N) ≅ Ext_{A}^{•}(M, N), M, N ∈ A/J–mod.

The other functors in (1) are defined as adjoints of i_{∗} and j^{∗}: (i^{∗}, i_{∗}, i^{!}) and (j_{!}, j^{∗}, j_{∗}) form adjoint triples, with j_{!}, j_{∗} full embeddings of derived categories. Thus, i^{∗} (resp., i^{!}) is left (resp., right) adjoint to i_{∗} and similarly for j^{∗}. More specifically, i^{∗} (resp., i^{!}) is realized as the left (resp. right) derived functor of the module functor i^{∗} : A–mod → A/J−mod (resp., i^{!} : A–mod → A/J−mod) which associates to any A-module M the largest quotient module (resp., submodule) which lies in the full subcategory A/J−mod of A–mod. Similarly, j_{∗} := ℝHom_{eAe}(eA, −) is the right derived functor of j_{∗} = Hom_{eAe}(eA, −) : eAe−mod → A–mod, while j_{!} := Ae ⊗_{eAe}^{𝕃} − is the left derived functor of j_{!} := Ae ⊗_{eAe} − : eAe−mod → A–mod. For later use, observe that j_{!} (resp., j_{∗}) is right (resp., left) exact. See [CPS1] and [P] for more details.

1In general, Ext_{A}^{i}(M, N) ≅ Hom_{D^{b}(A–mod)}(M, N[i]) for A-modules M, N. Here we regard A–mod as a subcategory of D^{b}(A–mod) consisting of complexes concentrated in degree 0, while N[i] is the complex which is 0 in all degrees except for N in degree −i.

The following result shows that the standard (resp., costandard) modules behave well with respect to the adjoint j_{!} (resp., j_{∗}). This result will be used in §3 below, but also has independent interest.

Lemma 3. For any ω ∈ Ω,

j_{!}Δ_{Ω}(ω) ≅ j_{!}Δ_{Ω}(ω) ≅ Δ(ω), j_{∗}∇_{Ω}(ω) ≅ j_{∗}∇_{Ω}(ω) ≅ ∇(ω).

Proof. For an eAe-module M,

j_{!}M ≅ Ae ⊗_{eAe}^{𝕃} M ∈ D^{b}(A–mod).

Thus, if M ∈ A–mod is projective, j_{!}M ≅ j_{!}M ≅ Ae ⊗_{eAe} M ∈ A–mod is projective since j_{!} ≅ Ae ⊗_{eAe} −. In particular, if ω ∈ Ω is maximal, then Δ_{Ω}(ω) is projective, so that j_{!}Δ_{Ω}ω ≅ j_{!}Δ_{Ω}(ω) ∈ A–mod is projective. Because j^{∗}L(λ) ≅ L_{Ω}(λ) provided λ ∈ Ω and equals 0 otherwise, we have

Hom_{A}(j_{!}Δ_{Ω}(ω), L(λ)) ≅ Hom_{eAe}(Δ_{Ω}(ω), j^{∗}L(λ)) ≠ 0

for any λ ∈ Λ if and only if λ = ω (in which case, this Hom-set is one-dimensional). It follows that j_{!}Δ_{Ω}(ω) is the projective indecomposable A-module corresponding to ω, i. e., j_{!}Δ_{Ω}(ω) ≅ Δ(ω), as required.

The opposite algebra A^{op} is also quasi-hereditary [CPS1, p. 93]—in fact, the axioms for a quasi-hereditary algebra are left-right symmetric. Using linear duality M ↦ M^{∗} to identify A^{op}−mod with the category of right A-modules, we see that A^{op}–mod is a highest weight category with weight poset Λ, standard objects ∇(λ)^{∗} and costandard objects Δ(λ)^{∗}. Let M ∈ eAe–mod. Since

Hom_{k}(Ae ⊗_{eAe} M, k) ≅ Hom_{eAe}(Ae, Hom_{k}(M, k)),

it follows from Lemma 1 and the fact that Ae has a Δ_{Ω}-filtration in the highest weight category mod–eAe of right eAe-modules that,

j_{!}M ≅ Ae ⊗_{eAe}^{𝕃} M ≅ Ae ⊗_{eAe} M ≅ j_{!}M,

for any M ∈ A–mod(Ω) which has a Δ_{Ω}-filtration. In particular, Tor_{1}^{eAe}(Ae, M) = 0.

Now suppose that ω ∈ Ω is such that j_{!}Δ_{Ω}(ω′) ≅ Δ(ω′) for any ω′ ∈ Ω satisfying ω′ > ω. Let P_{Ω}(ω) be the projective cover in A–mod(Ω) of Δ_{Ω}(ω) and form the exact sequence 0 → C(ω) → P_{Ω}(ω) → Δ_{Ω}(ω) → 0, so that C(ω) has a Δ-filtration in A–mod(Ω) with sections of the form Δ_{Ω}(ω′), ω′ > ω. Applying the left exact module functor j_{!} and using the previous paragraph, we obtain an exact sequence 0 → j_{!}C(ω) → j_{!}P_{Ω}(ω) → j_{!}Δ_{Ω}(ω) → 0 in A–mod. By our assumption on ω, j_{!}C(ω) has a Δ-filtration with sections of the form Δ(ω′), ω′ > ω, while j_{!}P_{Ω}(ω) is the projective cover of L(ω) (or of Δ(ω)).

Let Q(ω) be the kernel of the natural surjection P (ω) ↠ Δ(ω), and form the commutative diagram in A–mod

0  → j_{!}C(ω) → j_{!}P_{Ω}(ω) → j_{!}Δ_{Ω}(ω) →  0

[figure: σ; ∼; π]

0  → Q(ω) → P (ω) → Δ(ω) →  0,

with exact rows. In this diagram, π is the adjunction morphism j_{!}j^{∗}Δ(ω) ≅ j_{!}Δ_{Ω}(ω) → Δ(ω), while the morphism σ is induced by π. By the snake lemma, the kernel K of the surjective map π is isomorphic to the cokernel of the injective map σ. Because Δ_{Ω}(ω) ≅ j^{∗}Δ(ω), j^{∗}K = 0. Thus, the A-module K has the form K = i_{∗}K′ for some K′ ∈ A/J–mod. On the other hand, i^{∗}Q(ω) = 0: since i^{∗} is right exact, Q(ω) has a Δ-filtration by Δ(τ), τ ∈ Ω, and i^{∗}Δ(τ) = 0 for any such τ. (Note that i^{∗}M ≅ M/JM for any M.) Thus, Hom_{A}(Q(ω), K) = Hom_{A}(Q(ω), i_{∗}K′) ≅ Hom_{A/J}(i^{∗}Q(ω), K′) = 0. It follows that K = 0, showing that j_{!}Δ_{Ω}(ω) ≅ Δ(ω), as required.

This completes the proof that j_{!}Δ_{Ω}(ω) ≅ j_{!}Δ_{Ω}(ω) ≅ Δ(ω). The dual statement for ∇(ω) is obtained in a similar fashion. □

3. Tilting modules and the Ringel dual. We merely summarize a few key facts. For more details, see [R], [D3], [CPS5, §4].

Let A–mod be a highest weight category with weight poset (Λ, ≤). For λ ∈ Λ, let Y (λ) be the indecomposable tilting module of highest weight λ. Thus, Y (λ) has two filtrations: 0 = F^{0} ⊂ F^{1} ⊂ ⋯ ⊂ F^{m} = Y (λ) and Y (λ) = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{n} = 0 in which F^{i}/F^{i−1} ≅ Δ(λ_{i}), 0 < i ≤ m, (resp., G_{j−1}/G_{j} ≅ ∇(μ_{j}), 0 < j ≤ n) for some λ_{i}, μ_{j} ∈ (−∞, λ] = {μ ∈ Λ | μ ≤ λ} with λ_{1} = λ = μ_{1}. Necessarily, Δ(λ) (resp., ∇(λ)) appears with multiplicity 1 as a section in the filtration F^{•} (resp., G_{•}).

Fix a complete tilting module

Y = ⨁_{λ∈Λ} Y (λ)^{⊕r_{λ}}, (3)

which just means that each multiplicity r_{λ} is positive. The Ringel dual of A–mod is the category E–mod, where

E = End_{A}(Y). (4)

Although the algebra E depends on the choice of the positive integers r_{λ}, a different choice leads to a Morita equivalent algebra; it is in this sense that we speak of the Ringel dual of A–mod. The category E-mod is itself a highest weight category with weight poset (Λ, ≤^{op}), the opposite of the poset for A–mod. The standard modules Δ_{E}(λ) for E–mod can be described as follows in terms of the standard modules Δ(λ) for A–mod. Regard Y as a left E-module, and form the contravariant “tilting functor”

(5) T_{A} = Hom_{A}(−, Y) : A−mod ⟶ E−mod.

Then

(6) Δ_{E}(λ) ≅ T_{A}(Δ(λ)) := Hom_{A}(Δ(λ), Y).

In addition, Y is a full tilting module in E–mod and the double centralizer property End_{E}(T) ≅ A holds. In particular, A–mod identifies with the Ringel dual of the highest weight category E–mod.

Lemma 4. (a) Suppose M ∈ A–mod has a Δ-filtration 0 = F^{0} ⊂ F^{1} ⊂ ⋯ ⊂ F^{m} = M with sections F^{i}/F^{i−1} ≅ Δ(λ_{i}), λ_{i} ∈ Λ. Then

0 = T_{A}(M/F^{m}) ⊂ T_{A}(M/F^{m−1}) ⊂ ⋯ ⊂ T_{A}(M/F^{0}) = T_{A}M

forms a Δ-filtration of T_{A}M with sections

T_{A}(M/F^{i−1})/T_{A}(M/F^{i}) ≅ Δ_{E}(λ_{i}).

(b) For λ ∈ Λ, let P (λ) be the projective indecomposable A-module of highest weight λ. Then T_{A}P (λ) ≅ Y_{E}(λ), the indecomposable tilting module in E–mod of highest weight λ.

(c) For λ ∈ Λ, T_{A}Y (λ) ≅ P_{E}(λ), the projective indecomposable E-module of highest weight λ.

(d) If Proj_{A} denotes the full subcategory of A–mod whose objects are projective modules, then T_{E} ∘ T_{A}|_{Proj_{A}} ≅ Id_{Proj_{A}}, the identity functor on Proj_{A}. Similarly, if Tilt_{A} denotes the full subcategory of A–mod whose objects are tilting modules, then T_{E} ∘ T_{A}|_{Tilt_{A}} ≅ Id_{Tilt_{A}}.

Proof. Part (a) follows from Lemma 1 and (6). It is easy to see that T_{A}P (λ) is an indecomposable summand of Y as an E-module. Hence, T_{A}P (λ) is an indecomposable tilting module in E-mod. But P (λ) has a Δ-filtration in which the sections are all of the form Δ(μ), μ ≥ λ. In addition, Δ(λ) occurs exactly one time, namely, as the “top” section. Thus, by (a) and (6), T_{A}P (λ) has a Δ_{E}-filtration in which Δ_{E}(λ) is a section and λ is maximal among all μ in the opposite poset (Λ, ≤^{op}) for which Δ_{E}(μ) is a section of T_{A}P (λ). Thus, T_{A}P (λ) ≅ Y_{E}(λ), proving (b). The proof of (c) is similar; we leave the details to the reader. Part (d) also follows. □

We remark that the above result, in a slightly different form and in terms of basic algebras, already appears in [R, Thm. 6 and Lemma 7]. The difference is that [R] works with the covariant functor Hom_{A}(Y, −), while we have found it more convenient to cast things in terms of our contravariant functor (5).

Example 1. We assume the reader is familiar with the theory of q-Schur algebras. The special case of q = 1 (classical Schur algebras) will be discussed in §4 and the general case briefly touched upon in §5. We briefly indicate how the Ringel dual works in this setting. So let S_{q}(n, r) be the q-Schur algebra of bidegree (n, r) over an algebraically closed field k. Here 0 ≠ q ∈ k. When q = 1, we denote S_{q}(n, r) just by S(n, r); see §4. Also, S_{q}(n, r)–mod is a highest weight category with weight poset Λ^{+}(n, r), the set of partitions of r of length at most n. We have the following explicit description of the Ringel dual of S_{q}(n, r)–mod. This result is proved independently in [D4] and [DPS] (and in [D3] when q = 1, which will be the case of most interest in this paper).

Theorem 5. Let E_{q}(n, r)−mod be the Ringel dual of S_{q}(n, r). Then E_{q}(n, r)−mod is a highest weight category with poset Λ^{+}(n, r)′, the ideal in Λ^{+}(r) = Λ^{+}(r, r) consisting of all dual partitions λ′ for λ ∈ Λ^{+}(n, r).

Furthermore, we have the following explicit identifications of the Ringel dual:

(a) If n ≥ r, there is an equivalence of categories

G : E_{q}(n, r)−mod ⟶^{∼} S_{q}(n, r)−mod,
Δ_{E_{q}(n,r)}(λ) ↦ Δ(λ′).

(b) If n < r, there is an equivalence of categories

G : E_{q}(n, r)−mod ⟶^{∼} S_{q}(r, r)−mod[Λ^{+}(n, r)′],

Δ_{E_{q}(n,r)}(λ) ↦ Δ(λ′).

3. The main results

In this section, we work in the setting of reductive groups over an algebraically closed field k. Thus, our point of view here is closest to that in [D1].

Fix a connected, reductive algebraic group G over k with maximal torus T and Borel subgroup B ⊇ T; let H ⊇ T be a Levi subgroup of a parabolic subgroup P ⊇ B. For simplicity, we will assume that the derived group of G is simply connected; hence, that of H is also simply connected. We let X = Hom(T, 𝔾_{m}) denote the weight lattice of T and X^{∨} = Hom(𝔾_{m}, T) the dual lattice of coweights. The pairing ⟨ , ⟩ : X × X^{∨} → ℤ ≅ End(𝔾_{m}) is the usual pairing induced by composition.

The root groups of G relative to T are the minimal elements in the set of non-identity unipotent subgroups of G normalized by T. Each root group U_{α} is isomorphic to 𝔾_{a}; the conjugation action of T on U_{α} defines a unique root α ∈ X. Let Φ denote the root system of G, i. e. the set of roots relative to T. The root system Φ_{H} of H consists of those roots β ∈ Φ such that U_{β} ⊂ H. See [Sp] for more discussion.

The Borel subgroup B determines a unique positive system of roots, Φ^{+} = {α ∈ Φ | U_{α} ⊂ B }, together with its associated system of simple roots Π ⊂ Φ^{+}. Moreover, Φ_{H}^{+} = Φ^{+} ∩ Φ_{H} and Π_{H} = Π ∩ Φ_{H}, are the positive and simple systems of roots in Φ_{H} determined by the Borel subgroup B ∩ H of H. In turn, the positive systems define two orders ≤ and ≤_{H} on X: set ν ≤ μ (resp., ν ≤_{H} μ) if μ − λ ∈ ℤ^{+}Φ^{+} (resp., μ − ν ∈ ℤ^{+}Φ_{H}^{+}). We can also define the sets X^{+} (resp., X^{+(H)}) of dominant (resp., H−dominant) weights as those elements λ ∈ X which occur as the highest weights in an irreducible rational G−module L(λ) (resp., irreducible rational H–module L_{H}(λ)).

It will be useful to single out the dominant and H-dominant weights in any subset Λ of X. Thus, if Λ ⊆ X, we set Λ^{+} = Λ ∩ X^{+} and Λ^{+(H)} = Λ ∩ X^{+(H)}, respectively.

There is also a more “internal” description of the sets X^{+} and X^{+(H)}: for a simple root α, let α^{∨} : 𝔾_{m} → T be the corresponding coroot. Thus, α^{∨}(t), t ∈ k^{×}, denotes

the image of [ \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ] under a map of the group SL_{2}(k) which sends the root group

[ \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ] isomorphically to the root group U_{α} and sends the opposite root group of

SL_{2}(k) to the opposite root group U_{−α}. Then a weight ν ∈ X is dominant if and only if ⟨ν, α^{∨}⟩ ≥ 0 for each α ∈ Π. The weight ν belongs to X^{+(H)} if ⟨ν, α^{∨}⟩ ≥ 0 for all α ∈ Π_{H} = Π ∩ Φ_{H}^{+}. The equivalence of this internal notion of dominance with the “external” one involving irreducible modules is well-known for semisimple groups, and extends without difficulty to reductive groups. For more discussion, see [Jan3, II.2.6].

The following result follows easily from the fact that the Cartan matrix entry associated to distinct simple roots is always ≤ 0.

Lemma 6. Suppose μ ≤_{H} λ are weights in X, with μ ∈ X^{+(H)} and λ ∈ X^{+}. Then μ ∈ X^{+}. In other words, X^{+} is an ideal in the poset (X^{+(H)}, ≤_{H}).

Continuing with our notation, the categories of finite dimensional rational G and H modules will be denoted C and C_{H}, respectively. (Recall the convention at the start of Section 2.) Both C and C_{H} are highest weight categories with weight posets (X^{+}, ≤) and (X^{+(H)}, ≤_{H}), respectively. In the case of C, the standard module Δ(λ), λ ∈ X^{+}, is the Weyl module with highest weight λ, while the costandard module ∇(λ) is the induced module ind_{B^{−}}^{G}λ, where B^{−} is the Borel subgroup containing T and opposite to B and λ denotes the one-dimensional B^{−}-module defined by inflating the character λ from T to B^{−}. Similar definitions apply to C_{H}.

If Λ ⊆ X, it is useful (as suggested by previous notation) to write C[Λ] for the category of all modules M ∈ Ob(C) such that every composition factor L(ν) of M satisfies ν ∈ Λ. Obviously, ν ∈ Λ^{+}, so that C[Λ] = C[Λ^{+}]. Similarly, C_{H}[Λ] is defined and C_{H}[Λ] = C_{H}[Λ^{+(H)}].

If F ⊆ X, we denote by X_{F} the ideal in the poset (X, ≤) generated by F, i. e., X_{F} = { γ ∈ X | γ ≤ ϕ forsomeϕ ∈ F }. If we regard a subset Y ⊆ X as a subposet of X, then Y_{F} := X_{F} ∩ Y is an ideal in Y.

Now let Ω = ω+ ℤΦ_{H} be a coset of ℤΦ_{H} in X. If F ⊆ Ω, observe that Ω_{F} = { γ ∈ Ω | γ ≤_{H} ϕ forsomeϕ ∈ F }. Set Γ = X_{Ω}, and for F ⊆ Ω, observe that Γ_{F} = Γ_{Ω_{F}}.

Let F ⊆ Ω. At least when Γ_{F} is a finitely generated ideal in X (e. g., when F is finite), the category C[Γ_{F}] = C[Γ_{Ω_{F}}] is fully embedded in the full category of rational G-modules at the derived category level, cf. [CPS1, Thm. 3.9]. In particular, if M, N belong to C[Γ_{F}^{+}], then^{2}

Ext_{C[Γ_{F}^{+}]}^{•}(M, N) ≅ Ext_{G}^{•}(M, N).

(See also [D2].) Obviously, if F is finite, then the set Γ_{F}^{+} of dominant weights in Γ_{F} is also finite. In this way, the homological algebra of the category C[Γ] = C[Γ^{+}] is completely determined by the various categories C[Γ_{F}] = C[Γ_{F}^{+}], as F ranges over the finite subsets of Γ^{+}. Each C[Γ_{F}^{+}] is equivalent to the finite dimensional module category of a quasi-hereditary algebra A_{F} [CPS1, Thm. 3.6]. Note that Ω_{F}^{+} is a coideal of weights in Γ_{F}^{+}, yielding a recollement diagram

D^{b}(C[Γ_{F}^{+}\Ω_{F}^{+}]) \underleftarrow{i^{∗}} ; \underrightarrow{i_{∗}} ; \underleftarrow{i_{!}} D^{b}(C[Γ_{F}^{+}]) \underleftarrow{j_{!}} ; \underrightarrow{j^{∗}} ; \underleftarrow{j_{∗}} D^{b}(C(Ω_{F}^{+})). (7)

as in (1). Here C(Ω_{F}^{+}) denotes the quotient category C[Γ_{F}^{+}]/C[Γ_{F}^{+}\Ω_{F}^{+}]. As with (1), the above derived category functors are all induced from corresponding functors at the module category level, e. g., the quotient functor j^{∗} is induced by a quotient functor j^{∗} : C[Γ_{F}^{+}] → C(Ω_{F}^{+}).

Next, we consider the truncation functor

π_{Ω} : C[Γ^{+}] → C_{H}[Ω] = C_{H}[Ω^{+(H)}], M ↦ π_{Ω}M = ⨁_{τ∈Ω} M_{τ}. (8)

Here M_{τ}, τ ∈ Ω, denotes the τ-weight space of the rational G-module M.

The following property was observed in [D1] at the character-theoretic level, and at the module level in the irreducible case. Compare also [Sm]. Multiplicity results like Prop. 3.2(c) below seem to appear first in [Jan2] in a semisimple Lie algebra context. See especially Theorem 1.18 of that reference, attributed to Jantzen and Deodhar independently. The treatment in [Jan2] formalizes induction, but does not formalize truncation, as in [D1], though parts of the proof may be viewed as implicitly involving truncation.

Proposition 7. Fix λ, ν, μ ∈ Γ^{+}.

(a) If λ ∈ Ω^{+}, then

π_{Ω}Δ(λ) ≅ Δ_{H}(λ), π_{Ω}∇(λ) ≅ ∇_{H}(λ), π_{Ω}L(λ) ≅ L_{H}(λ).

(b) If ν ∉ Ω^{+}, then

π_{Ω}Δ(ν) ≅ π_{Ω}∇(ν) ≅ π_{Ω}L(ν) = 0.

(c) If μ ∈ Ω^{+}, then the multiplicity of L(μ) as a composition factor of Δ(λ) (resp., ∇(λ)) equals the multiplicity of L_{H}(μ) as a composition factor of Δ_{H}(λ) (resp., ∇_{H}(λ)).

Ext_{C}^{•}(M, N) ≅ Ext_{G}^{•}(M, N),

2This isomorphism easily implies that, given finite dimensional G-modules M, N, we have where the former group is formed using n-fold Yoneda extensions.

Proof. To establish (a), we first observe that for L(λ) and ∇(λ), λ ∈ Ω^{+}, it is already proved in [Jan3, II.2.11] that π_{Ω}L(λ) ≅ L_{H}(λ) and π_{Ω}∇(λ) ≅ ∇_{H}(λ). This easily implies the required assertion for Δ(λ), λ ∈ Ω. In fact, since Δ(λ) (resp., Δ_{H}(λ)) and ∇(λ) (resp., ∇_{H}(λ)) have the same characters, π_{Ω}Δ(λ) is a rational H−module with the same character and high weight as Δ_{H}(λ).^{3} Clearly, π_{Ω}Δ(λ) is generated as an H–module by a vector v of weight λ. (It is even generated by such a vector over the hyperalgebra of the negative Borel subgroup of H.) Thus, the map Δ_{H}(λ) → π_{Ω}Δ(λ) obtained from the well-known universal property of Δ_{H}(λ), sending a high weight vector of Δ_{H}(λ) to v is an isomorphism. This proves (a).

If λ, ν ∈ Γ^{+}\Ω^{+}, then ν < τ for some τ ∈ Ω. So τ −ν must be a sum of positive roots, one of which is a simple root β ∉ Φ_{H}. Consequently, if θ < ν, then θ ∉ Ω, because the difference τ − θ must still involve a positive multiple of β. So π_{Ω}Δ(ν), π_{Ω}∇(ν), π_{Ω}L(ν) must all be zero, and (b) holds.

Finally, observe that if L(μ) is a composition factor of either Δ(λ) or ∇(λ), then μ ∈ Γ^{+}. Thus (c) follows from exactness of π_{Ω}, together with (a) and (b). □

Before stating the next result, we draw several conclusions about the various posets involved. First, for any subset F ⊆ Ω, Ω_{F}^{+} is a coideal in the poset (Γ_{F}^{+}, ≤). In turn, Γ_{F}^{+} is an ideal in (X^{+}, ≤). Suppose now that F is finite. Thus, the highest weight category C(Ω_{F}) := C[Γ_{F}^{+}]/C[Γ_{F}^{+}\Ω_{F}^{+}] with weight poset (Ω_{F}^{+}, ≤) = (Ω_{F}^{+}, ≤_{H}) is defined [CPS1, Thm. 3.5]. On the other hand, Lemma 6 implies that Ω_{F}^{+} is an ideal in the poset (X^{+(H)}, ≤_{H}). Thus, the highest weight category C_{H}[Ω_{F}^{+}] with weight poset (Ω_{F}^{+}, ≤_{H}) = (Ω_{F}^{+}, ≤) is defined. Similarly, Ω^{+} is a coideal in (Γ^{+}, ≤), but an ideal in (X^{+(H)}, ≤_{H}). We form the quotient category C(Ω^{+}) := C[Γ^{+}]/C[Γ^{+}\Ω^{+}]; see the discussion on quotient categories given in [F, Ch. 15]. Then the irreducible objects in the categories C(Ω^{+}) and C_{H}[Ω^{+}] are both indexed by the same poset (Ω^{+}, ≤) = (Ω^{+}, ≤_{H}). The category C_{H}[Ω^{+}] is a highest weight category, and we will prove that C(Ω^{+}) is equivalent to it.^{4}

Theorem 8. If F ⊆ Ω^{+} is finite, the functor π_{Ω^{+}}, restricted to the category C[Γ_{F}^{+}], factors through the quotient morphism j^{∗} : C[Γ_{F}^{+}] → C(Ω_{F}^{+}) to produce an equivalence

C(Ω_{F}^{+})\overline{π_{Ω}} ; →_{∼} C_{H}[Ω_{F}^{+}] (9)

3Alternatively, without appealing to [Jan3], this fact follows formally from the observation that the character of Δ(λ) is the alternating sum over w ∈ W (the Weyl group) of the characters of the Verma modules (for the complex Lie algebra having the same root type as G) of high weight w · 0. Similar remarks apply to Δ_{H}(λ). Taken with the remainder of this paragraph (and the dual argument), the results cited in [Jan3] can be reproved.

4In case Ω^{+} and Γ^{+} are finite, the discussion of [CPS1] implies that C(Ω^{+}) is a highest weight category. This case is close to the situation discussed in the first paragraph of §2.2 for a highest weight category A–mod. In general, the finite generation hypotheses of [CPS1, Thm. 3.9] need not hold, so it is a priori unclear if the quotient category C(Ω^{+}) is a highest weight category.

of categories. Similarly, the functor π_{Ω}, restricted to the subcategory C[Γ^{+}], factors through the quotient morphism j^{∗} : C[Γ^{+}] → C(Ω^{+}) to produce an equivalence

C(Ω^{+})\overline{π_{Ω}} ; →_{∼} C_{H}[Ω^{+}] (10)

of categories. In particular, C_{H}(Ω^{+}) is a highest weight category.

Proof. First assume that (9) has been proved. Taking directed unions, as F ranges through all finite subsets of Ω^{+}, we also obtain an equivalence

\overline{π_{Ω}} : C(Ω^{+})\underrightarrow{∼}C_{H}[Ω^{+}],

so that (10) holds.

Thus, it suffices to prove (9). Since F is finite, C_{H}[Ω_{F}^{+}] is a highest weight category with finitely many simple objects, the irreducible modules L_{H}(ν) with ν ∈ Ω_{F}^{+}. Of course, C(Ω_{F}^{+}) is a highest weight category with the same finite weight poset, and we also note that its standard and costandard objects are inherited from C[Γ_{F}^{+}]. Clearly, the functor π_{Ω}, applied to the category C[Γ_{F}^{+}], factors through the quotient category C(Ω_{F}^{+}), as may be observed by its behavior on irreducible objects. Thus, we obtain a commutative diagram

C[Γ_{F}^{+}]  \overline{j^{∗}}  \overline{}C(Ω_{F}^{+})

[figure: π_{Ω}; \overline{π_{Ω}}]

C_{H}[Ω_{F}^{+}]  .

The quotient morphism j^{∗} admits a left adjoint j_{!} and a right adjoint j_{∗}, as discussed in §2.2. Both j_{!} and j_{∗} are sections for j^{∗}, while j_{!} is right exact and j_{∗} is left exact. Since

\overline{π_{Ω}} = \overline{π_{Ω}}id_{C(Ω_{F}^{+})}
= \overline{π_{Ω}}j^{∗}j_{!}
= π_{Ω}j_{!},

the exactness of π_{Ω} and the right exactness of j_{!} imply that \overline{π_{Ω}} is right exact. A similar calculation shows that \overline{π_{Ω}} is left exact. Therefore, \overline{π_{Ω}} is an exact functor. But now, using Prop. 7, the Comparison Theorem 2 applies, giving an equivalence

{\overline{π_{Ω}} : C(Ω_{F}^{+}) \atop ∼→ C_{H}[Ω_{F}^{+}].}

□

It often turns out that Ω^{+} is itself finite. The proof of the proposition below relies on the fact that each entry in the inverse of the Cartan matrix of a simple algebraic group is a positive rational number. For a nice exposition of this fact, see [LT].

Proposition 9. Assume that H is a Levi subgroup of a proper parabolic subgroup P of the reductive group G. Assume that no simple component of Φ_{H} is a component of Φ. (For example, this last condition holds if G, or its commutator subgroup G′, is simple.) If Ω = ω + ℤΦ_{H} is a coset of ℤΦ_{H} in X, then the set Ω^{+} is finite. In this case, Γ^{+} is also finite.

Proof. The proof easily reduces to the case when Φ is connected and G = G′. We give the argument in this case. We may assume Ω^{+} is non-empty and choose the coset representative ω to be a dominant weight. Since H is proper, there is a simple root α in Π \ Π_{H}. Write ω as a rational linear combination of simple roots, and let a denote the coefficient of α in this expression. Since any other element μ of Ω differs from ω by an integral linear combination of the roots in Π_{H}, the coefficient of α in the analogous expression for μ is also equal to a.

Let {ϖ_{α}}_{α∈Π} ⊂ X be the fundamental dominant weights corresponding to Φ, i. e., ⟨ϖ_{α}, β^{∨}⟩ = δ_{α,β}, α, β ∈ Π. If we assume μ = ∑_{β∈Π} m_{β}β is dominant, then we have

a = ∑_{β∈Π} m_{β}r_{β,α},

where m_{β} ≥ 0 is an integer and the coefficients r_{β,α} are positive rational numbers. This equation clearly has only finitely many solutions. Thus, Ω^{+} is finite.

Given γ ∈ Γ^{+}, γ ≤ ω for some ω ∈ Ω. Since Ω is W_{H}-stable, ω ≤ ω′ ∈ Ω^{+}. Therefore, γ ≤ ω′ ∈ Ω^{+}. The finiteness of Ω^{+} implies that Γ^{+} is finite. □

The category equivalences of the theorem, when coupled with the derived category recollement diagrams for the various C(Ω_{F}^{+}), are quite powerful. We have the following Ext-transfer result, using Lemma 3:

Corollary 10. If λ, μ are dominant weights in Ω, then

Ext_{G}^{•}(Δ(λ), L(μ)) ≅ Ext_{H}^{•}(Δ_{H}(λ), L_{H}(μ))

and

Ext_{G}^{•}(L(μ), ∇(λ)) ≅ Ext_{H}^{•}(L_{H}(μ), ∇_{H}(λ)).

Proof. We just prove the first assertion; the second follows similarly. Let λ, μ ∈ Ω^{+}. Choose some finite subset F ⊂ Ω such that λ, μ ∈ Ω_{F}^{+}. (It is not difficult to see that one could work with F = Ω.) By Lemma 3,

Δ(λ) ≅ j_{!}Δ_{Ω_{F}^{+}}(λ) = j_{!}Δ_{Ω_{F}^{+}}(λ),

using the recollement diagram (7). Thus, by Thm. 8,

Ext_{G}^{•}(Δ(λ), L(λ)) \begin{matrix}≅ Ext_{C[Γ_{F}^{+}]}^{•}(Δ(λ), L(μ)) \\ ≅ Hom_{D^{b}(C[Γ_{F}^{+}])}^{•}(Δ(λ), L(μ)) \\ ≅ Hom_{D^{b}(C[Γ_{F}^{+}])}^{•}(j_{!}Δ_{Ω_{F}^{+}}(λ), L(μ)) \\ ≅ Hom_{D^{b}(C(Ω_{F}^{+}))}^{•}(Δ_{Ω_{F}^{+}}(λ), j^{∗}L(μ)) \\ ≅ Hom_{D^{b}(C(Ω_{F}^{+}))}^{•}(Δ_{Ω_{F}^{+}}(λ), L_{Ω_{F}^{+}}(μ)) \\ ≅ Hom_{D^{b}(C[Ω_{F}^{+}])}^{•}(\overline{π_{Ω}}Δ_{Ω_{F}^{+}}(λ), \overline{π_{Ω}}L_{Ω_{F}^{+}}(μ)) \\ ≅ Ext_{H}^{•}(Δ_{H}(λ), L_{H}(μ)), \end{matrix}

as required. □

Remarks 11. (a) If p ≥ h (the Coxeter number of G) and the modular Lusztig conjecture holds for G, then the dimensions of the groups Ext_{G}^{n}(Δ(λ), L(μ)) and Ext_{G}^{n}(L(μ), ∇(λ)) can be determined, at least when λ, μ are p-regular, in terms of the coefficients of Kazhdan-Lusztig polynomials for the affine Weyl group W_{p} associated to G [A1, (2.1.2)]. From these dimensions, the dimensions of the groups Ext_{G}^{n}(L(λ), L(μ)) can also be determined, using [CPS3, (3.9.1)]. In some cases, when p is too small or λ, μ are not p-regular, Ext-calculations can be reduced, using the above Ext-transfer, to similar calculations for Levi subgroups H where the above hypotheses are satisfied. An illustration of this kind of procedure, in type A, is given in [PS2, §6].

(b) As pointed out to us by one referee, one can also establish Cor. 10 directly. We sketch the details. Let P^{−} = H ⋉ U_{P^{−}} be the parabolic subgroup with Levi factor H which contains the negative Borel subgroup B^{−}. Then Ext_{G}^{•}(Ł(μ), ∇(λ)) ≅ Ext_{G}^{•}(L(μ), Ind_{P^{−}}^{G}∇_{H}(λ)) ≅ Ext_{P^{−}}^{•}(L(μ), ∇_{H}(λ)), where ∇_{H}(λ) is regarded as a rational P^{−}-module through inflation. In this expression, the second isomorphism follows easily from [CPSK, Thm. 3.1], applied to the module denoted there by M = ∇_{H}(λ). (One needs that R^{n}Ind_{P^{−}}^{G}∇_{H}(λ) = 0 for n > 0, but this follows immediately from the Grothendieck spectral sequence corresponding to the functor factorization Ind_{B^{−}}^{G} = Ind_{P^{−}}^{G} ∘ Ind_{B^{−}}^{P^{−}} together with Kempf’s theorem [Jan3, II.4.6] for G and P^{−}.) Consider the Hochschild-Serre spectral sequence

E_{2}^{p,q} = H^{p}(H, H^{q}(U, L(μ)^{∗} ⊗ ∇_{H}(λ))) ⇒ Ext_{P^{−}}^{p+q}(L(μ), ∇_{H}(λ)).

As a rational H-module, L(μ) ≅ L_{H}(μ) ⊕ D, where the composition factors L_{H}(ξ) of D satisfy ξ ∉ Ω as well as ξ < μ. If q > 0, the weights of T in H^{q}(U, L_{H}(μ)^{∗} ⊗ ∇_{H}(λ))) have the form ξ + θ, where ξ is a non-trivial sum of roots in Φ^{+}\Φ_{H}^{+} and θ lies in ℤΦ_{H}. Hence, H^{p}(H, H^{q}(U, L_{H}(μ)^{∗} ⊗ ∇_{H}(λ)) = 0 if q > 0. (Here we are using the fact that the weights in H^{q}(U, k), q > 0, are all non-trivial sums of roots in Φ^{+}\Φ_{H}^{+}; see [CPSK, §2].) Next, observe that for all q ≥ 0, H^{p}(H, H^{q}(U, D ⊗∇_{H}(λ))) = 0. Otherwise, there exists weights ζ, ξ, and σ with the following properties: (1) ζ is a non-trivial sum of positive roots, not all in Φ_{H}^{+}; (2) ξ is a sum, possibly empty, of roots in Φ^{+}\Φ_{H}^{+}; (3) σ ∈ ℤΦ_{H}; and (4) ξ + (−μ + ζ) + (λ − σ) ∈ ℤΦ_{H}. (Here −μ + ζ corresponds to a weight in D and λ − σ to a weight in ∇_{H}(λ).) We use the fact that, given a rational H-module M, if H^{q}(H, M) ≠ 0, then some weight of M must lie in ℤΦ_{H} [CPSK, Thm. 2.4]. Equation (4) is clearly impossible since λ−μ ∈ ℤΦ_{H}. Finally, these facts together imply that the spectral sequence collapses to the required Ext_{H}^{•}(L_{H}(μ), ∇_{H}(λ)).

Although the above argument is more direct, the original proof of Cor. 10, based on the categorical equivalence of Thm. 8, remains more conceptual. In addition, it should be mentioned that the existence of Thm. 8 led to the discovery of Cor. 10.

(c) Anton Cox has pointed out to us a result of Donkin and Erdmann, given in [E1, (4.3)], which is similar to the first part of Cor. 10, but with L(μ) replaced with Δ(μ). Such a result is in the spirit of the original results in [C], especially [C, Cor. 6.7]. Specifically, the result states, under the same hypotheses as in Cor. 10, that

Ext_{G}^{•}(Δ(λ), Δ(μ)) ≅ Ext_{H}^{•}(Δ_{H}(λ), Δ_{H}(μ)).

Using the methods of this paper, a proof of this result is obtained from that for Cor. 10 by replacing L(μ) by Δ(μ) and using the fact, mentioned in (2), that shows j^{∗}Δ(μ) ≅ Δ_{H}(μ). Dually, one has

Ext_{G}^{•}(∇(λ), ∇(μ)) ≅ Ext_{H}^{•}(∇_{H}(λ), ∇_{H}(μ)).

As in the case of Cor. 10 itself, a direct proof of this isomorphism can be based on the argument given in (b); this is, in fact, similar to the argument in [E1].

(d) We finally observe that Thm. 8 trivially holds when k is a field of characteristic 0. In this case, all the highest weight categories involved are semisimple. One referee has raised the question of the validity of the theorem for reductive group schemes over ℤ or over the localization ℤ_{(p)}. This seems likely to us, but we have not pursued it.

Using an Euler characteristic argument, Cor. 10 gives an interesting interpretation of the multiplicity identification [D1]. See also [Jan3, II.5.21].

Corollary 12. Let λ, μ be dominant weights in Ω. Then

[Δ(λ) : L(μ)] = [Δ_{H}(λ) : L_{H}(μ)].

Proof. If M ∈ C, let [M] denote its image in the Grothendieck group of C. Use the same notation to denote objects in the Grothendieck group of C_{H}.

For ν, τ ∈ X^{+}, write

[L(ν)] = ∑_{σ} p_{σ,ν}[Δ(σ)]. (11)

This is certainly possible, using only the finitely many σ ≤ ν, by a well-known argument with unitriangular matrices. Also, p_{ν,ν} = 1. By a formal result (“Delorme’s theorem”— see [CPS3, (3.2)]),

p_{σ,ν} = ∑_{i}(−1)^{i} dim Ext_{G}^{i}(L(ν), ∇(σ)). (12)

Similarly, we have

[L_{H}(ν)] = ∑_{σ} p_{σ,ν}^{H}[Δ_{H}(σ)], (13)

where

p_{σ,ν}^{H} = ∑_{i}(−1)^{i} dim Ext_{H}^{i}(L_{H}(ν), ∇_{H}(σ)).

The previous Cor. 10 shows that if σ, ν both belong to Ω, then

p_{σ,ν} = p_{σ,ν}^{H}. (14)

Write

\begin{aligned}[Δ(λ)] & = ∑_{τ} q_{τ,λ}[L(τ)]; \\ [Δ_{H}(λ)] & = ∑_{ξ} q_{ξ,λ}^{H}[L_{H}(ξ)]. \end{aligned} (15)

We must show that q_{μ,λ} = q_{μ,λ}^{H}. But if we substitute (15) into (11), we find that the q’s are, as expected, “inverse” to the p’s:

\begin{aligned}∑_{ν} p_{τ,ν}q_{ν,λ} & = δ_{τ,λ} \\ ∑_{ν} p_{τ,ν}^{H}q_{ν,λ}^{H} & = δ_{τ,λ}. \end{aligned}

Now (14) easily yields, recursively, the desired equalities. □

4. Schur algebras, I: first Ext-transfer.

In this section, we situate the theory of Schur algebras in the setting of the previous section.

We first recall some notation, some of it already used. For a positive integer r, let Λ^{+}(r) be the set of all partitions λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ 0) of r. It will be convenient to regard λ as an infinite non-increasing sequence of non-negative integers, eventually equal to 0. Given λ ∈ Λ^{+}(r), the dual partition λ′ is defined by setting λ′_{i} = #{λ_{j} | λ_{j} ≥ i}. For λ, μ ∈ Λ^{+}(r), put λ ⊴ μ provided that λ_{1} + ⋯ + λ_{i} ≤ μ_{1} + ⋯ + μ_{i} for all i. In this way, Λ^{+}(r) is a poset. For a positive integer n, the subset Λ^{+}(n, r) of Λ^{+}(r) consisting of all partitions λ with at most n nonzero parts is a coideal in Λ^{+}(r). Clearly, λ ⊴ μ ⟺ μ′ ⊴ λ′.

Given λ ∈ Λ^{+}(r_{1}) and μ ∈ Λ^{+}(r_{2}), define λ + μ ∈ Λ^{+}(r_{1} + r_{2}) by setting (λ + μ)_{i} = λ_{i} + μ_{i} for all i.

Let k be an algebraically closed field of characteristic p ≥ 0. Put V = k^{n}, the space of n × 1 column vectors, and, given a positive integer r, let V^{⊗r} be the space of r-tensors. The symmetric group 𝔖_{r} of degree r acts on V^{⊗r} on the right by place displacement: (v_{1} ⊗ ⋯ ⊗ v_{r})σ = v_{σ(1)} ⊗ ⋯ ⊗ v_{σ(1)}, for v_{1}, ⋯, v_{n} ∈ V and σ ∈ 𝔖_{r}. The Schur algebra S(n, r) is defined to be the endomorphism algebra End_{k𝔖_{r}}(V^{⊗r}) for this action of 𝔖_{r} on V^{⊗r}. Because the action of GL_{n}(k) on V extends to an action of GL_{n}(k) on V^{⊗r}, there is a natural homomorphism ξ : GL_{n}(k) → GL(V^{⊗r}). This (left) action of GL_{n}(k) commutes with the action of 𝔖_{r}, and S(n, r) identifies with the subalgebra of End(V^{⊗r}) generated by the image of ξ. For further details on the general theory of Schur algebras, see [Gr, Ch. 2]. Other references on Schur algebras are [D4] and [P].

It is well-known that the module category S(n, r)–mod is a highest weight category with respect to the poset Λ^{+}(n, r) described above. We can also identify S(n, r)–mod as the category of polynomial representations of GL_{n}(k) which are homogeneous of degree r. In addition, for M, N ∈ S(n, r)–mod,

Ext_{S(n,r)}^{•}(M, N) ≅ Ext_{GL_{n}(k)}^{•}(M, N). (16)

(See [D2] or the discussion in §3.)

Consider the reductive group GL_{r}(k). For 0 < n < r, form the Levi subgroup

H = GL_{n}(k) × 𝔾_{m}^{r−n} of GL_{r}(k). Explicitly, H consists of matrices [ \begin{matrix}A & 0 \\ 0 & t \end{matrix} ] with

A ∈ GL_{n}(k), t ∈ 𝔾_{m}^{r−n}.^{5} The character group X of the maximal torus T of diagonal matrices identifies naturally with sequences λ = (λ_{1}, ⋯, λ_{r}) of integers; we call |λ| := λ_{1} + ⋯ + λ_{r} the degree of λ. The root system Φ of T on Lie(GL_{r}(k)) consists of those sequences of weight 0 with exactly two nonzero entries ±1. Then α = (a_{1}, ⋯, a_{r}) ∈ Φ is positive if the first nonzero a_{i} equals +1. For λ, μ ∈ X, put λ ≥ μ provided λ − μ is a sum of positive roots. If λ, μ ∈ Λ^{+}(r), then λ ≤ μ ⟺ λ ⊴ μ. Also, α ∈ Φ belongs to the root system Φ_{H} of T in Lie(H) if and only if a_{i} = 0 for i > n.

The irreducible GL_{r}(k)-modules are indexed by the subset X^{+} of X consisting of non-increasing sequences λ = (λ_{1}, λ_{2}, ⋯, λ_{r}). In turn, the irreducible H-modules are indexed by the set X^{+(H)} ⊂ X of sequences (μ_{1}, ⋯, μ_{n}, ⋯, μ_{r}) of integers in which μ_{1} ≥ μ_{2} ≥ ⋯ ≥ μ_{n}.

Let ω = (r, 0, ⋯, 0) ∈ X^{+(H)} ∩ X^{+} and form the coset Ω = ω + ℤΦ_{H}. Let F = {ω}. Then Ω_{F}^{+} = Λ^{+}(n, r) is an ideal in X^{+(H)} and a coideal in Γ_{F}^{+} = Λ^{+}(r). The category C[Γ_{F}^{+}] is equivalent to the category S(r, r)–mod. It is well-known that S(n, r)–mod is the quotient category of S(r, r) by the Serre subcategory S(r, r)–mod[Λ^{+}(r)\Λ^{+}(n, r)]. Thus, Cor. 10 identifies S(n, r)-mod with the highest weight category C_{H}[Ω_{F}^{+}] associated to the Levi subgroup H.

In this case, the Ext-identifications in Cor. 10 coincide with those obtainable from the more direct quotient category viewpoint for S(n, r); see [PS2, Thm. 6.1], which in turn was based on Lemma 3. However, we can obtain more information, as we now describe.

Theorem 13. Let λ, μ ∈ Λ^{+}(r). Suppose that λ̃, μ̃ are obtained from λ, μ by one of the following processes (or any iteration of them):

(i) Removal or addition of equal first rows; (ii) Removal or addition of equal last rows, when the total number of nonzero rows

5In the proof of Thm. 13, we will also use the Levi subgroup 𝔾_{m}^{r−n} × GL_{n}(k) consisting of matrices [ \begin{matrix}t & 0 \\ 0 & A \end{matrix} ], A ∈ GL_{n}(k), t ∈ 𝔾_{m}^{r−n}. The reader can immediately modify the discussion below to include this case.

is the same; (iii) Removal or addition of equal first columns; (iv) Removal or addition of equal last columns, when the total number of nonzero

columns is the same. Here we have identified the partitions with their Young diagrams in the usual way.

Let r̃ denote the common degree |λ̃| = |μ̃|. Then

Ext_{S(r,r)}^{•}(Δ(λ), L(μ)) ≅ Ext_{S(r̃,r̃)}^{•}(Δ(λ̃), L(μ̃))
Ext_{S(r,r)}^{•}(L(λ), ∇(μ)) ≅ Ext_{S(r̃,r̃)}^{•}(L(λ̃), ∇(μ̃)).

Proof. Notice that part (iv) follows from part (i), and part (ii) follows from part (iii). Also, it is enough in each case to treat only the “removal” alternative. For part (iii), let λ̃, μ̃ be obtained by removal of an equal first column from λ, μ. Then Δ(λ̃), L(μ̃) have the same restrictions to SL_{n}(k) as do Δ(λ), L(λ), respectively, where n is the length of the common first column of λ, μ. However, if M, N are homogeneous modules of GL_{n}(k) with the same degree of homogeneity, then Ext_{GL_{n}(k)}^{n}(M, N) ≅ Ext_{SL_{n}(k)}^{n}(M, N). Thus, using (16),

Ext_{S(n,r)}^{•}(Δ(λ), L(μ)) ≅ Ext_{SL_{n}(k)}^{•}(Δ(λ), L(μ))

and a similar statement holds for r̃ in place of r (even if n > r̃). The desired result now follows from Cor. 10 or [PS2, Thm. 6.1].

For Part (i), first note that λ, μ each have at most m = 1 + r̃ parts. Then we have, by using (16) again,

Ext_{S(r,r)}^{•}(Δ(λ), L(μ)) ≅ Ext_{S(m,r)}^{•}(Δ(λ), L(μ)) ≅ Ext_{GL_{m}(k)}^{•}(Δ(λ), L(μ)).

This is a slight abuse of notation, since the meaning of Δ(λ), L(μ) remains the same, though the objects themselves and categories they are in varies. Let G = GL_{m}(k), and let H in Cor. 10 be the group 𝔾_{m}×GLr̃(k), and set H̃ ≅ GLr̃(k) equal to the subgroup

of all invertible matrices [ \begin{matrix}1 & 0 \\ 0 & A \end{matrix} ] where A is an r̃×r̃ matrix. Clearly, λ−μ = λ̃−μ̃ as

weights for the full diagonal subgroup of GL_{m}(k). We denote the common difference by η. If η is not in the root lattice for GL_{n}(k), then

Ext_{G}^{•}(Δ(η), L(μ)) ≅ Ext_{H}^{•}(Δ_{H}(λ), L_{H}(μ)) = 0.

However, if η is in this root lattice, it is also in the root lattice for H̃, and, thus, of H. Hence, Cor. 10 applies, and we have

Ext_{G}^{•}(Δ(λ), L(μ)) ≅ Ext_{H}^{•}(Δ_{H}(λ), L_{H}(μ)).

However, it is obvious that Δ_{H}(λ), L_{H}(μ) are obtained from their counterparts, ΔH̃(λ̃), LH̃(μ̃) on H̃ by just tensoring (externally) with a 1-dimensional module in which 𝔾_{m} acts through the homomorphism t ↦ t^{r−r̃}. Part (iii) is now a consequence of, say, the Künneth formula.

An alternative proof of part (ii) may be obtained by similarly replacing H = 𝔾_{m} ×H̃ with the Levi subgroup (in the notation above the statement of the theorem) H̃ × 𝔾_{m}, with r − r̃ here equal to the common length of the bottom rows of λ, μ. (This is how part (ii), and, then, part (iv) were discovered.) □

Remarks 14. (a) Similar results hold for decomposition numbers, as observed in [D1], for cases (i) and (iii). (Case (ii) works as well for decomposition numbers.) That is, if λ, μ are related to λ̃, μ̃ as hypothesized, then

[Δ(λ) : L(μ)] = [Δ(λ̃) : L(μ̃)].

(b) In the spirit of Remark 11(b), we have, using the hypotheses and notation of Thm. 13,

Ext_{S(r,r)}^{•}(Δ(λ), Δ(μ)) ≅ Ext_{S(r̃,r̃)}^{•}(Δ(λ̃), Δ(μ̃))
Ext_{S(r,r)}^{•}(∇(λ), ∇(μ)) ≅ Ext_{S(r̃,r̃)}^{•}(∇(λ̃), ∇(μ̃)).

The proof is an evident modification of that of Thm. 13.

5. Schur algebras, II: Second Ext-transfer.

We will be considering Schur algebras S(n, r) for various choices of positive integers n, r. We assume that k has positive characteristic p. 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), the schemetheoretic kernel of 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. Let ρ_{n} = (n − 1, n−2, ⋯, 1, 0) ∈ Λ^{+}(n, \frac{n(n−1)}{2}) and let St = L((p−1)ρ_{n}) ∈ S(n, (p−1)\frac{n(n−1)}{2})–mod be the Steinberg module. As a module for the infinitesimal subgroup G_{1}, St is projective, irreducible and self-dual. For λ ∈ Λ^{+}(n, r), let, as usual, P (λ) ∈ S(n, r)–mod be the indecomposable projective module with head L(λ).

The following lemma is well-known, and follows easily from the Steinberg tensor product theorem and the Hochschild-Serre spectral sequence for the pair (GL_{r}(k), G_{1}). (For example, see [Jan3, Prop. II.3.19] for a proof of (a).) Recall from §3 that, given λ ∈ Λ^{+}(r, r), Y (λ) denotes the indecomposable tilting module for S(r, r) of high weight λ.

Lemma 15. Let λ ∈ Λ^{+}(r, r) and set s = pr + (p − 1)\frac{r(r−1)}{2}. Then:

(a) Δ(λ)^{(1)} ⊗ St ≅ Δ(pλ + (p − 1)ρ_{r}) ∈ S(r, s)−mod;

(b) P (λ)^{(1)} ⊗ St ≅ P (pλ + (p − 1)ρ_{r}) ∈ S(r, s)−mod.

(c) Y (λ)^{(1)} ⊗ St ≅ Y (pλ + (p − 1)ρ_{r}) ∈ S(r, s)−mod.

Define a function

d : Λ^{+}(r) → Λ^{+}(r, s), λ ↦ (pλ′ + (p − 1)ρ_{r})′. (17)

We will consider the left exact, additive functor

𝔍 : S(r, r)−mod → S(s, s)−mod[Λ^{+}(r, s)′] (18) defined as the composition

𝔍 = G_{2} ∘ T_{2} ∘ F_{1} ∘ G_{1} ∘ T_{1}

of five functors given as follows. First, T_{1} = T_{S(r,r)} : S(r, r)−mod → E(r, r)−mod is the tilting functor (5), while G_{1} : E(r, r)−mod → S(r, r)−mod is the category equivalence described in Thm. 5(a) when q = 1. Next, F_{1} : S(r, r)−mod → S(r, s)−mod is the exact functor defined by F_{1}(M) = M^{(1)} ⊗ St for all M ∈ S(r, r)−mod, while T_{2} is the tilting functor T_{S(r,s)}, and G_{2} is the category equivalence given in Thm. 5(b) for E_{q}(r, s), when q = 1. For λ ∈ Λ^{+}(r), we have

𝔍Δ(λ) ≅ Δ(d(λ)), 𝔍P (λ) ≅ P (d(λ)), (19)

where Δ(d(λ)) and P (d(λ)) are standard and indecomposable projective modules in the highest weight category S(s, s)−mod[Λ(r, s)′]. (Observe that Δ(d(λ)) is a standard module in S(s, s)–mod, but P (d(λ)) is not necessarily a projective module in S(s, s)– mod.)

Suppose that Γ is an ideal in the poset Λ^{+}(a, b). In the following proof, we will use the fact that

[Δ(λ) : L(μ)] = [P (μ) : Δ(λ)], ∀λ, μ ∈ Γ

in the category S(a, b)−mod[Γ]. This follows from Brauer-Humphreys reciprocity [CPS3, (3.11)], plus the fact that S(a, b)−mod[Γ] admits a strong duality D.^{6}

Lemma 16. The image of d : Λ^{+}(r) → Λ^{+}(s) is a union of blocks for the highest weight category S(s, s)−mod[Λ^{+}(r, s)′].

Proof. Suppose that d(λ) and μ ∈ Λ^{+}(r, s)′ belong to the same block for the category S(s, s)−mod[Λ^{+}(r, s)′]. (That is, L(d(λ)) and L(μ) belong to the same block as explained in §2.) We can assume that there is a nonzero Ext^{1}-between the irreducible modules L(d(λ)) and L(μ).

\underline{Case 1}. Suppose that d(λ) ▷ μ, so that, by the universal mapping property of standard modules, 0 ≠ [Δ(d(λ)) : L(μ)] = [P (μ) : Δ(d(λ))]. In terms of the category S(r, s)– mod, this means that [Y (μ′) : Δ(d(λ)′)] ≠ 0. Hence, d(λ)′ and μ′ belong to the same block for the weight poset Λ^{+}(r, s) of S(r, s)–mod. Now we must show that μ′ = d(τ)′ for some τ ∈ Λ^{+}(r). Again, we can reduce to the case where there is a nonzero Ext^{1} between L(d(λ)′) and L(μ′) in S(r, s)–mod. Using Lemma 15, we see that either 0 ≠ [Δ(d(λ)′) : L(μ′)] = [Δ(λ′)^{(1)} ⊗ St : L(μ′)] (i. e., d(λ)′ ▷ μ′) or 0 ≠ [P (d(λ)′) : Δ(μ′)] = [P (λ′)^{(1)} ⊗ St : Δ(μ′)] (i. e., μ′ ▷ d(λ)′). In either case, the exactness of the functor F_{1} implies that μ′ = pτ′ + (p − 1)ρ_{r} for some τ ∈ Λ^{+}(r), as required.

\underline{Case 2.} Now suppose that d(λ) ◁ μ. Then we can assume that 0 ≠ [Δ(μ) : L(d(λ))] = [P (d(λ)) : Δ(μ)] in S(s, s)−mod[Λ(r, s)′]. This means that, back in S(r, s)–mod, we have [Y (d(λ)′) : Δ(μ′)] ≠ 0. Thus, d(λ)′ and μ′ belong to the same block in Λ^{+}(r, s) for S(r, s)–mod. Then repeating the argument as in Case 1, we conclude that μ = d(τ) for some τ ∈ Λ^{+}(r). □

6More precisely, D : S(a, b)−mod → S(a, b)−mod is a contravariant equivalence such that D^{2} ≅ Id and DL(λ) ≅ L(λ) for all λ ∈ Λ^{+}(a, b).

The highest weight category S(s, s)−mod[Λ^{+}(r, s)′] has the form A–mod for a quasihereditary algebra A ≅ S(s, s)/J for some idempotent ideal J in S(s, s). Let B(r) be the product of the blocks of A whose irreducible modules have the form L(d(λ)) for some λ ∈ Λ^{+}(r). Thus, B(r) = eAe for a central idempotent e ∈ A. The functor 𝔍 : S(r, r)−mod → S(s, s)−mod[Λ(r, s)′] induces a functor 𝔍 : S(r, r)−mod → B(r)– mod. We thank Alison Parker for pointing out an error in the original version of the following result.

Theorem 17. The functor 𝔍 induces an equivalence 𝔍′ : S(r, r)−mod → B(r)−mod of categories. In particular, S(r, r) is Morita equivalent to the algebra B(r). The category B(r)−mod is a highest weight category with poset Λ^{+}(r) which isomorphic to the subposet d(Λ^{+}(r)) of Λ^{+}(s). The equivalence 𝔍′ maps standard, costandard and simple objects indexed by λ ∈ Λ^{+}(r) to the corresponding standard, costandard and simple objects in B(r)−mod indexed by d(λ).

Proof. If P (λ) is the projective indecomposable cover of Δ(λ) in S(r, r)–mod, then 𝔍P (λ) ≅ P (d(λ)) is the projective indecomposable cover of Δ(d(λ)) in B(r)–mod, and Lemma 16 implies that all projective indecomposable B(r)-modules arise this way. Furthermore, using Lemma 4(d) and the definition of the functor 𝔍, we have

Hom_{S(r,r)}(P(λ), P(μ)) ≅ Hom_{B(r)}(𝔍P(λ), 𝔍P(μ))

for all λ, μ ∈ Λ^{+}(r). Now

P = ⨁_{λ∈Λ^{+}(r)} P (λ), 𝔍P = ⨁_{λ∈Λ^{+}(r)} P (d(λ))

are projective generators for S(r, r)–mod and B(r)–mod, respectively. We have A := End_{S(r,r)}(P) is isomorphic to B := End_{B(r)}(𝔍P), while S(r, r)–mod is equivalent to Amod and B(r)–mod is equivalent to B–mod. Thus, S(r, r)–mod and B(r)–mod are equivalent by an equivalence 𝔍′ agreeing with 𝔍 on any object P (λ).

Observe that, given λ, μ ∈ Λ^{+}(r), λ ⊴ μ if and only if d(λ) ⊴ d(μ). Also, by (19), 𝔍′ maps Δ(λ) to Δ(d(λ)). Hence, 𝔍′L(λ) ≅ L(d(λ)). It follows that 𝔍′ maps the injective envelope I(λ) of L(λ) to the injective injective envelope I(d(λ)) of L(d(λ)) in B(r)– mod. Since ∇(λ) (resp., ∇(d(λ))) is the largest submodule of I(λ) (resp., I(d(λ))) all of whose composition factors L(μ) satisfy μ ⊴ λ (resp., μ ⊴ d(λ)), it follows that 𝔍′∇(λ) ≅ ∇(d(λ)). □

As a consequence, we obtain the following cohomological result.

Corollary 18. (a) Let M, N ∈ S(r, r)−mod. Then

Ext_{S(r,r)}^{•}(M, N) ≅ Ext_{S(s,s)}^{•}(𝔍′M, 𝔍′N).

(b) Let W, Z ∈ {L, Δ, ∇, Y }. Then for λ, μ ∈ Λ^{+}(r), we have

Ext_{S(r,r)}^{•}(W (λ), Z(μ)) ≅ Ext_{S(s,s)}^{•}(W (d(λ)), Z(d(μ))).

Proof. The equivalence 𝔍′ induces a full embedding D^{b}(S(r, r)−mod) → D^{b}(B(r)−mod) of bounded derived categories. Because B(r)–mod is a direct product of blocks of S(s, s)−mod[Λ^{+}(r, s)′], there is another full embedding

D^{b}(B(r)−mod) → D^{b}(S(s, s)−mod[Λ^{+}(r, s)′])

of derived categories. On the other hand, Λ^{+}(r, s)′ is an ideal in Λ^{+}(s), so that the recollement diagram (1) provides another full embedding

D^{b}(S(s, s)−mod[Λ^{+}(r, s)′]) → D^{b}(S(s, s)−mod).

Thus, we obtain by composition a full embedding D^{b}(S(r, r)−mod) → D^{b}(S(s, s)−mod). Since these full embeddings are induced by exact functors on respective module categories, assertion (a) is clear from (2) (and the accompanying footnote).

Finally, (b) follows from (a) since, by construction, 𝔍′W (λ) ≅ W (d(λ)) for all W ∈ {L, Δ, ∇, Y }. For example, 𝔍′Y (λ) is the indecomposable tilting module in B(r)–mod of highest weight d(λ). Since it is indecomposable, it remains the indecomposable tilting module in S(s, s)−mod[Λ^{+}(r, s)′] of highest weight λ. However, the functor i_{∗} : S(s, s)−mod[Λ^{+}(r, s)′] → S(s, s)–mod takes indecomposable tilting modules to indecomposable tilting modules. Therefore, 𝔍′Y (λ) identifies with Y (d(λ)) as an S(s, s)-module. □

Remark 1. Let a be a positive integer. In the above discussion, the functor 𝔍′ : S(r, r)−mod → S(s, s)−mod[Λ^{+}(r, s)′] can be replaced by a functor

𝔍′_{a} : S(r, r)−mod → S(s_{a}, s_{a})−mod[Λ^{+}(r, s_{a})′],

where s_{a} = p^{a}r + (p^{a} − 1)\frac{r(r−1)}{2}. It is defined in the same way as 𝔍′, but replacing the “twisting” functor F_{1} : S(r, r)−mod → S(r, pr + (p − 1)\frac{p(p−1)}{2}) by the functor

F_{a} : S(r, r)−mod → S(r, p^{a}r + (p^{a} − 1)\frac{r(r − 1)}{2}), M ↦ M^{(r)} ⊗ St_{a},

where M^{(r)} denotes the rational GL_{r}(k)-module obtained by making g ∈ GL_{r}(k) act through F^{a}(g), and St_{a} denotes the Steinberg module of highest weight (p^{a} −1)ρ_{r}. The straightforward modifications in the proofs are left to the reader.

6. Variations on the theme

In this section, we mainly discuss (very briefly) how some of the previous results work in a quantized situation. Throughout this section, k will be a fixed algebraically closed field of characteristic p ≥ 0. Let ζ ∈ k be a primitive ℓth root of unity. Some further restrictions will be placed on ℓ; these will be discussed below at the appropriate place. We view k as an algebra over the ring Z := ℤ[v, v^{−1}] of Laurent polynomials, under the map Z → k, v ↦ ζ.

1. Quantum groups at a root of unity. Because we want to consider quantum enveloping algebras over fields of positive characteristic, we will closely follow the discussion given by Andersen-Wen [AW]. Let A = [a_{i,j}] be an n × n (classical) Cartan matrix corresponding to a finite root system Φ. Fix a set Π = {α_{1}, ⋯, α_{n}} of simple roots for Φ; hence, a_{i,j} = ⟨α_{i}, α_{j}^{∨}⟩. Let X be the lattice of integral weights for Φ and let X^{+} be the corresponding set of dominant weights.

Now let U be the quantum enveloping algebra over Z corresponding to A; U has generators E_{i}^{(m)}, F_{i}^{(m)}, K_{i}^{±}, 1 ≤ i ≤ n, m ≥ 0, and forms a Z-form of the quantum algebra over ℚ(v) with generators E_{i}, F_{i}, K_{i}^{±} satisfying the quantum Serre relations which we do not repeat here. We let U^{0} be the subalgebra of U generated by the K_{i}^{±} and the

[ \begin{matrix}K_{i}; c \\ t \end{matrix} ] = ∏_{i=1}^{t} \frac{K_{i}v^{c−s+1} − K_{i}^{−1}v^{−v+s−1}}{v^{s} − v^{−s}}

c ∈ ℤ, t ∈ ℤ_{≥0}.

Put U_{k} := k ⊗_{Z} U, and let C = U_{k}–mod be the category of finite dimensional integrable type 1 U_{k}-modules. Then using standard results of [APW1],[APW2], [AW], in particular, the quantum version of Kempf’s theorem^{7}, it follows that U_{k}–mod is a highest weight category with weight poset Λ = X^{+}. An argument is provided in [DS] for a somewhat weaker result—entirely adequate for our purposes here—that the modules in C form a “k-finite highest weight category.”

Given I ⊆ {1, ⋯, n}, let X^{+(I)} ⊆ X be the set of weights λ ∈ X satisfying ⟨λ, α_{i}^{∨}⟩ ≥ 0 for all i ∈ I. Let U_{Ik} = k ⊗ U_{I}, where U_{I} is the “Levi” subalgebra of U generated by the E_{i}^{(m)}, F_{i}^{(m)}, for i ∈ I and 1 ≤ j ≤ n, together with U_{k}^{0}. Let C_{I} = U_{Ik}-mod be the category of integrable, type 1 modules for the quantum algebra U_{Ik}. It is a highest weight category with weight poset X^{+(I)}. For λ ∈ X^{+(I)}, let Δ_{I}(λ), ∇_{I}(λ), L_{I}(λ) denote the corresponding standard, costandard, and irreducible objects.

Now the main results of §3 carry through with little but notational change: Fix a coset Ω = ω + ℤI of ℤI in X. For any finite subset F ⊂ Ω^{+}, there is an equivalence of categories C(Ω_{F}^{+}) ≅ C_{I}[Ω_{F}^{+}]. In particular, if λ, μ ∈ Ω^{+}, then

Ext_{U_{k}}^{•}(Δ(λ), L(μ)) ≅ Ext_{U_{Ik}}^{•}(Δ_{I}(λ), L_{I}(μ))

and

Ext_{U_{k}}^{•}(L(μ), ∇(λ)) ≅ Ext_{U_{Ik}}^{•}(L_{I}(μ), ∇_{I}(λ)).

We remark that when k has characteristic 0, the dimensions of Ext_{C}^{i}(Δ(λ), L(μ)) and Ext_{C}^{i}(L(μ), ∇(λ))) can be computed in terms of Kazhdan-Lusztig polynomials, using the fact that, thanks to [KL] and [KT], C often has a Kazhdan-Lusztig theory in the sense of [CPS3]. See also [T]. The latter survey notes some restrictions on ℓ are required when Φ is not of type A—e. g., ℓ must be unequal to 2 for type D_{2n+1}, n > 1, and E_{8} requires ℓ be at least 25. (But the analogous restrictions in the non-simply laced cases are not known explicitly.) Strictly speaking, the results in [T] only give character formulas. However, these results are sufficient in the framework of [CPS4] to establish homological results, in particular a “Kazhdan-Lusztig theory”, in the presence of suitable “pre-Hecke operators.” In the quantum case, these can be constructed when ℓ ≥ h (the Coxeter number) using wall-crossing functors. Now [KL] provides an equivalence (with the mentioned restrictions on ℓ) of the finite dimensional modules in the quantum ℓth root of unity case with the full category of all finite length modules for an affine Lie algebra which have only certain composition factors (all of level −ℓ − h), and which are direct sums of weight spaces. This makes the requisite pre-Hecke operators available for the latter category, at least when ℓ ≫ h. We conjecture they are then available for ℓ > 1 (with other restrictions on ℓ as above, for types different from type A) by virtue of Kumar’s translation functors [Ku].

7This is proved in [APW1], [AW] with some restrictions on ℓ; however, these restrictions are completely removed in [RH].

2. The special case of q-Schur algebras. We maintain the notation of the previous subsection, and put q = ζ^{2}. The q-Schur algebras S_{q}(n, r) arise in the special case of type A_{n−1}. Explicitly, let Ṽ be a free Z-module of rank r. Let H̃ be the Hecke algebra over Z corresponding to the symmetric group 𝔖_{r}. There is a natural right action of H̃ on Ṽ^{⊗r} and S̃_{q}(n, r) := EndH̃(Ṽ^{⊗r}) [DD, (3.1.5)]. Over k, we just set S_{q}(n, r) := k ⊗ S̃_{q}(n, r). Then S_{q}(n, r)–mod is a highest weight category for weight poset λ^{+}(n, r). Now all the results of §4 hold in the q-Schur algebra case. They can be proved using either the quantum algebra U_{k} of type A_{n−1} discussed above or some version of the quantum general linear group [DD], [PW]. We leave further details to the interested reader.

Finally, we leave as an interesting open question the issue of extending the results of §5 to the case of the q-Schur algebras. In relation to [Le] and this question, see [PS2, (6.6b)].

3. Other variations. Most recently, we have been attracted to results such as those in section 3 for their potential application in the setting of characteristic p affine Lie algebras. We believe that the results do carry over, with an appropriate formalism, but prefer to leave such a development to a future project. Similarly, we believe that results very similar to those described in the present paper hold in the G_{1}T setting of [C].

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Department of Mathematics, University of Oklahoma, Norman, OK 73019 USA (Cline) Department of Mathematics, University of Virginia, Charlottesville, VA USA 22903 (Parshall, Scott)

Added December 31, 2004: A result similar in substance to Theorem 17 in the present paper, but formulated as a Morita equivalence of the Schur algebra S(r, r) with a quotient of a symmetric group algebra, localized at an idempotent, has been independently obtained by W.B. Turner (“Rock Blocks,” preprint (2004), Thm. 64) using finite group representation theory. In other words, S(r, r) is Morita equivalent to an algebra e(k𝔖_{s}/I)e, for some ideal I ◁ k𝔖_{s} and some idempotent e ∈ k𝔖_{s}. In the context of this paper, this result can be very briefly explained as follows:

One way to get from the Schur algebra categories of Thm. 17 to the localized symmetric group quotient is to apply the Schur functor to a projective generator, then (working over a suitable DVR) use Lemma 9.1 of our RIMS Kyoto 35 (1999) paper, Generic and q-rational representation theory, pp. 31-90. This latter result shows that the projective generator for the category 𝔍′S(r, r)–mod becomes a projective module for the quotient k𝔖_{s}/I. In fact, as Turner implicitly observes, one can replace k𝔖_{s}/I by a quotient of a single block B of k𝔖_{s}, since all partitions d(λ)′, λ ∈ Λ^{+}(r), have the same p-core.

As is well known, all Schur algebras S(n, r) with n ≥ r are Morita equivalent to Schur algebras S(r, r), while those with n < r are Morita equivalent to algebras fS(r, r)f with f an idempotent. Thus, Turner’s version of Thm. 17 shows all Schur algebras are Morita equivalent to algebras of the form e(k𝔖_{s}/I)e. In the special case n < p, a result of Erdmann is noted (in a quantum setting) in [DPS; Thm. 8.4] which shows that the Ringel dual of S(n, r) is actually Morita equivalent to a quotient of a symmetric group algebra, with no localization at an idempotent required. While we observe in [DPS] that Erdmann’s result holds in the “quantum case”—that is, for q-Schur algebras and Hecke algebras, at a root of unity—we do not know if there is a similar equivalence as in the above paragraphs for a general q-Schur algebra.