ROOTS OF UNITY AND POSITIVE CHARACTERISTIC

WOLFGANG SOERGEL

Abstract. This article gives an overview of [AJS94]. I outline our results to the effect that representation categories of both quantum groups at a root of one and semisimple algebraic groups in positive characteristic admit a common combinatorial model over ¡b¿Z¡/b¿. This allows in particular to deduce Lusztig’s old modular conjectures from their newer quantum analogues (which are a theorem).

1. Introduction

In [AJS94] we proved that for a fixed root system Lusztig’s quantum conjecture [Lus89, Lus90a] implies his modular conjecture [Lus80] in almost every characteristic, i.e. up to finitely many exceptions. For any given characteristic however we still do not know whether it is an exception or not. This article is meant as a guide to [AJS94]. The idea is to just cite the results of some heavy computations there thus allowing the main stream of arguments to flow without obstructions. We start explaining the problem in the easiest example.

Let k be an algebraically closed field of characteristic chark = p > 0 and consider the Lie algebra 𝔤 = 𝔰𝔩(n, k) for some n ≥ 2. A representation of 𝔤 is a pair (ρ, V) where V is a k-vectorspace an ρ : 𝔤 → End_{k}V a Lie algebra homomorphism. Such a representation is called restricted if and only if ρ(D^{p}) = ρ(D)^{p} for all D ∈ 𝔤. (Remark that D ∈ 𝔤 implies D^{p} ∈ 𝔤, since tr(D^{p}) = (trD)^{p}. We will from now on write D^{[p]} instead of D^{p} in order to avoid confusion with the p-th power in the enveloping algebra later on.) The problem is to describe the simple restricted representations of 𝔤 in some detail, for example to compute their dimensions.

More generally let G ⊃ B ⊃ T be a semisimple simply connected algebraic group over k, a Borel subgroup and a maximal torus. Let 𝔤 ⊃ 𝔟 ⊃ 𝔥 be the corresponding Lie algebras. By definition 𝔤 is the Lie algebra of all left invariant vector fields D on G, i.e. the Lie algebra of all k-linear derivations D of the ring k[G] of regular functions on the group that commute with left translations. Now obviously D ∈ 𝔤 implies D^{p} ∈ 𝔤 where we just take the p-th power in End_{k}k[G]. In order to avoid confusion later we write here again D^{[p]} instead of D^{p}.

A restricted representation of 𝔤 again is a pair (ρ, V) such that ρ(D^{[p]}) = ρ(D)^{p} for all D ∈ 𝔤. Every rational representation of G differentiates to a restricted representation of 𝔤. It is not true in general that simple representations of G differentiate to simple representations of 𝔤. Nevertheless a detailed knowledge of the simple restricted representations of 𝔤 leads via known results ([Jan87], II.3) to a detailed knowledge of all simple rational representations of G, and those in turn lead to a knowledge of all simple representations of finite Chevalley groups in their own characteristic, see e.g. [Car63].

Thus it is an important problem to describe the simple restricted representations of 𝔤. In our work we deduce such a description (for almost every characteristic) from similar results in the theory of quantum groups at a root of unity. This is the last step of a bigger program proposed by Lusztig to establish his conjectures describing the simple restricted 𝔤-modules. The previous steps from affine flag manifolds to Kac-Moody algebras to quantum groups at a root of unity have been accomplished by Kashiwara-Tanisaki [KT94] and Kazhdan-Lusztig [KL93, KL94, Lus94], so that by now the whole program is sucessfully completed in the simply laced cases, see [And94, Soe]. To settle the non simply laced case it remains only to extend [KT94] to non-integral weights.

2. Review of classical results

2.1. Verma modules and simple modules. We begin with an exposition of the classical theory of restricted representations of 𝔤. More details can be found in [Jan87], II.9.

Let U(𝔤) be the enveloping algebra of 𝔤. Define the restricted enveloping algebra U^{res}(𝔤) = U(𝔤)/(D^{[p]} − D^{p}) to be its quotient by the twosided ideal generated by the D^{[p]} − D^{p} for D ∈ 𝔤. Here we have D^{[p]} ∈ 𝔤 whereas D^{p} is the p-th power in U(𝔤). By definition a restricted representation of 𝔤 is nothing else but a U^{res}(𝔤)-module, and U^{res}(𝔤) obviously is a finite dimensional k-algebra (whose dimension can be seen to be p^{dim 𝔤}).

Let X = X(T) = Hom(T, k^{×}) be the weight lattice. Any ν ∈ X differentiates to give ν̄ ∈ 𝔥^{∗}. Remark that ν̄ = μ̄ if and only if ν − μ ∈ pX. Let kν̄ be the onedimensional representation of 𝔥 corresponding to ν̄. Now there is unique Lie algebra homomorphism 𝔟 ↠ 𝔥 left inverse to the inclusion 𝔥 ↪ 𝔟, thus we can consider the kν̄ also as representations of 𝔟.

One then induces to obtain the so-called “Baby-Verma-modules” Z(ν̄) = U^{res}(𝔤) ⊗_{U(𝔟)} kν̄. (The terminology is due to Cline, Parshall, and Scott. Indeed these modules are quite little, namely of finite dimension. In their behaviour however they turn out to be more difficult to control than ordinary Verma modules.) The Baby-Vermas have unique simple quotients L(ν̄), and the map ν ↦ L(ν̄) induces a bijection from X/pX to the isomorphism classes of simple restricted representations of 𝔤. The Baby-Verma-modules are well understood, so one might think the whole problem is to compute the Jordan-Hölder multiplicities [Z(λ̄) : L(μ̄)] for λ̄, μ̄ ∈ X/pX.

However the square matrix of these multiplicities has determinant zero in general, so it does not allow us to translate our knowledge about the Baby-Verma-modules into knowledge about their simple quotients. To overcome this difficulty we will consider G_{1}T -modules rather than restricted 𝔤-modules. Namely take the adjoint action of T on 𝔤. It induces an X-grading on 𝔟, 𝔤, U(𝔟), U(𝔤), U^{res}(𝔤) and it seems natural to consider the category U^{res}(𝔤)-mod_{X} of all X-graded U^{res}(𝔤)-modules M = ⨁_{ν∈X} M_{ν}. For any X-graded object M and any μ ∈ X define the shifted X-graded object M[μ] to have homogeneous parts (M[μ])_{ν} = M_{ν−μ}.

It turns out that it is even more natural to consider the full subcategory

C = C_{k} = \{ \begin{matrix}& M is finitely generated and \\ M ∈ U^{res}(𝔤)-mod_{X} & Hm = ν̄(H)m \\ & ∀H ∈ 𝔥, ν ∈ X, m ∈ M_{ν}. \end{matrix} \}

One can show that the category of all finitely generated X-graded U^{res}(𝔤)-modules breaks up into a direct sum

U^{res}(𝔤)-mod_{X}^{fg} = ⨁_{μ} C[μ]

with μ running over (a system of representatives of) the classes X/pX. The category C can also be identified with the category of finite dimensional representations of the subgroup scheme G_{1}T ⊂ G generated by T and the kernel G_{1} of the Frobenius. This will have no importance for us. However remark that every finite dimensional rational G-module E restricts to an object of C: We just differentiate the G-action to get a 𝔤-action and take the X-grading coming from the action of T.

If M resp. N are X-graded right resp. left modules over an X-graded ring R, then M ⊗_{R} N admits a natural X-grading with (M ⊗_{R} N)_{λ} the image of ⨁_{ν+μ=λ}(M_{ν} ⊗_{ℤ} N_{μ}) in M ⊗_{R} N. For ν ∈ X we denote by k_{ν} the U(𝔟)-module kν̄ considered as an X-graded object concentrated in degree ν and induce to get the X-graded U(𝔤)-module

Z(ν) = Z_{k}(ν) = U^{res}(𝔤) ⊗_{U(𝔟)} k_{ν}.

In this way we associate to every ν ∈ X the Baby-Verma-module Z(ν) = Z_{k}(ν) ∈ C. This comes with a distinguished basis vector v_{ν} = 1 ⊗ 1 of Z(ν)_{ν}. Our Z(ν) has a unique simple quotient L(ν) = L_{k}(ν) ∈ C. If we forget the X-grading on L(ν), Z(ν) we find our old L(ν̄), Z(ν̄).

The map ν ↦ L(ν) determines a bijection between X and the simple isomorphism classes in C. (This is the real reason why it is better to consider C rather than all of U^{res}(𝔤)-mod_{X}.) The basic problem we want to solve can now be formulated precisely.

Problem 2.1.1. Compute dim_{k} L(λ)_{ν} for any λ, ν ∈ X.

Consider the group ring ℤ[X] of the weight lattice. Each ν ∈ X gives an element e^{ν} ∈ ℤ[X]. For M ∈ C we form its character chM = ∑_{ν}(dim_{k} M_{ν})e^{ν} ∈ ℤ[X]. In this language our problem is to compute the characters chL(λ) of the simple objects.

The characters of our Baby-Verma-modules are easy to compute. One gets

chZ(λ) = ue^{λ}

with

u = ∏_{α∈R^{+}}(e^{0} + e^{−α} + e^{−2α} + … + e^{−(p−1)α}),

where R^{+} ⊂ X are the roots of B. Let us denote for M ∈ C by [M : L(λ)] ∈ ℕ the Jordan-Hölder multiplicity in C. Then we have [Z(λ) : L(λ)] = 1 and [Z(λ) : L(μ)] ≠ 0 implies μ ∈ λ − ℕR^{+}. Now the obvious equality chZ(λ) = ∑_{μ}[Z(λ) : L(μ)]chL(μ) tells us that our problem can be reformulated as

Problem 2.1.2. Compute [Z(λ) : L(μ)] for any λ, μ ∈ X.

2.2. The linkage principle. ([Jan87], II.9.12.) The so-called linkage principle says that lots of these multiplicities are zero a priori. More precisely let us recall the affine Weyl group W. It is an affine reflection group acting on X ⊗_{ℤ} ℝ, the semidirect product of the finite Weyl group W with the translations by the root lattice ℤR. For any l ∈ ℕ_{≥1} we define a new action of W on X by conjugating the old action first with a homothety by l and then with a translation by the halfsum of positive roots ρ = \frac{1}{2} ∑_{α∈R^{+}} α. In formulas, we set

w·_{l}λ = lw(l^{−1}(λ + ρ)) − ρ.

Recall p was the characteristic of k.

Theorem 2.2.1. [Linkage principle] [Z(λ) : L(μ)] ≠ 0 implies W ·_{p} λ = W ·_{p} μ, i.e. λ and μ are in the same orbit under the (·_{p})-action of the affine Weyl group W on X.

This already greatly reduces the number of multiplicities we actually have to compute. Using the linkage principle we arrive at the translation principle which further reduces the number of multiplicities we need to compute. This however needs some preparation.

2.3. Duality. First we discuss the duality on our category C = C_{k}. For any object M ∈ C the contragredient representation M^{∗} = Hom_{k}(M, k) with its X-grading given by (M^{∗})_{ν} = (M_{−ν})^{∗} certainly is again an object of C. This duality has the drawback not to fix the isomorphism classes of simple objects. To remedy this inconvenience, we consider a Chevalley antiautomorphism τ : 𝔤 → 𝔤, i.e. an involutive antiautomorphism that is the identity on 𝔥 and exchanges the root spaces 𝔤_{α} and 𝔤_{−α} for any root α. Then define M^{τ} ∈ C to be the vector space M^{∗} with the 𝔤-action given by (Df)(m) = f(τ(D)m) for all D ∈ 𝔤, f ∈ M^{∗}, m ∈ M and equip M^{τ} with the X-grading given by (M^{τ})_{ν} = (M_{ν})^{∗}. Now M ↦ M^{τ} is a contravariant exact involutive functor C → C, and L(λ)^{τ} ≅ L(λ) for all λ ∈ X. We often write Z(λ)^{τ} = Z^{τ}(λ).

It is a very useful fact in the theory that Z^{τ}(λ) is isomorphic to a “Z-object for the opposite Borel”. In particular Z^{τ}(λ) has a unique simple quotient, Z(λ) has a unique simple submodule, and these two are isomorphic to the same simple object, call it L(λ̌).

Lemma 2.3.1. We have

Ext_{C}^{i}(Z(λ), Z^{τ}(μ)) ≅ \{ \begin{matrix}k & i = 0, λ = μ; \\ 0 & else, \end{matrix}

Ext_{C}^{i}(Z^{τ}(λ), Z(μ)) ≅ \{ \begin{matrix}k & i = 0, λ = μ; \\ 0 & else. \end{matrix}

Proof. We only prove and use the most easy cases i = 0, 1. For the general case see for example [Jan87], II.11.1. Note that the second statement is the first for the opposite Borel. So we only treat the first statement. Suppose we are given M ∈ C such that M_{ν} = 0 if ν ∈ λ + ℕR^{+} and ν ≠ λ. Then by definition of Z(λ) we have an isomorphism Hom_{C}(Z(λ), M) → M_{λ} given by f ↦ f(v_{λ}), and in particular Ext_{C}^{1}(Z(λ), M) = 0. Thus the first statement is proven for μ ∉ λ+ℕR^{+} and for μ = λ. In the remaining case μ ∈ λ+ℕR^{+}, μ ≠ λ we use duality to identify Ext_{C}^{i}(Z(λ), Z^{τ}(μ)) = Ext_{C}^{i}(Z(μ), Z^{τ}(λ)) and thus reduce to a known case.

Remark 2.3.2. This proof also shows that Ext_{C}^{1}(Z(λ), Z(μ)) ≠ 0 implies μ ∈ λ + ℕR^{+} and μ ≠ λ.

2.4. The reciprocity principle. Every simple module for a finite dimensional algebra admits a projective cover, i.e. an indecomposable projective surjecting onto it, and this projective cover is unique up to nonunique isomorphism. Copying the arguments one sees that the same holds for our category C. Let me describe the known properties of the projective cover Q(λ) of a simple object L(λ).

First of all Q(λ) admits a so-called Z-filtration, i.e. a filtration where all sucessive quotients are isomorphic to some Z(μ). The multiplicity (Q(λ) : Z(μ)) of Z(μ) occuring as a subquotient is independent of the particular Z-filtration. This is deduced most easily from the fact that the chZ(λ) are linearly independent. Humphreys [Hum71] showed in this situation the following reciprocity principle:

Theorem 2.4.1. (Q(λ) : Z(μ)) = [Z(μ) : L(λ)].

Proof. For any module Q ∈ C with a Z-filtration we have (Q : Z(μ)) = dim_{k} Hom_{C}(Q, Z^{τ}(μ)) as is easily deduced from Lemma 2.3.1 by induction on the length of the Z-filtration. On the other hand we have for any module Z ∈ C the equaltities [Z : L(λ)] = [Z^{τ} : L(λ)] = dim_{k} Hom_{C}(Q(λ), Z^{τ}), as is easily deduced by induction on the length of Z.

The Q(λ) are known [Jan87], II.11.5(3) to be selfdual, Q(λ) ≅ Q(λ)^{τ}. It follows that they are not only the projective covers of the L(λ), but their injective hulls as well. It follows furthermore that the Q(λ) admit a Z^{τ}-filtration. For later use we cite from [Jan87], II.11.8 a specially easy case.

Lemma 2.4.2. Suppose λ ∈ pX − ρ. Then Q(λ) = Z(λ) = Z^{τ}(λ) = L(λ).

2.5. Alcove geometry. We want next to explain the translation principle and have to repeat some alcove geometry, see [Bou81]. Let E be a real euclidean space. A group W of orthogonal transformations of E is called a “reflection group” if and only if (1) W is generated by the reflections it contains and (2) the family H = {E^{s} | s ∈ W is a reflection} of reflecting hyperplanes is locally finite.

Since H is locally finite, the complement of the union of all reflecting hyperplanes E − ⋃_{H∈H} H is open. The connected components of this complement are called alcoves. Let A denote the set of all alcoves. Then for any alcove A ∈ A the map W → A, w ↦ wA is a bijection. In other words, the alcoves form a principal homogeneous W-space. Furthermore if A is an alcove, any x ∈ E is conjugate under W to exactly one point y in the closure of A. In other words, Ā is a fundamental domain for the action of W on E.

More generally remark that each hyperplane H ⊂ E defines a partition of E into three subsets, namely H itself and two open halfspaces. Consider the smallest partition of E that refines these partitions for every reflecting hyperplane H ∈ H. The parts of this partition are called “facets”. All facets are convex. The open facets are just the alcoves. The closure of any facet is a union of facets.

The facets which are open in some reflecting hyperplane are called walls. Let S denote the set of W-orbits of walls. The walls in the closure of any given alcove form a system of representatives for S. Any wall lies in the closure of precisely two alcoves. For A ∈ A and s ∈ S we denote by As ∈ A the unique alcove ≠ A sharing with A a wall from s.

A point v ∈ E is called special if (1) it is a facet by itself and (2) for every H ∈ H there exists H′ ∈ H which is parallel to H and contains v. We let P denote the set of special points.

All the notations introduced above will be applied to the reflection groups (W·_{p}) acting on the affine space E = X ⊗_{ℤ} ℝ (equipped with any W -invariant metric). Its reflecting hyperplanes are given by the equations H_{α,m} = {λ | ⟨λ + ρ, α^{∨}⟩ = pm} for α ∈ R, m ∈ ℤ. The special points are P = pX − ρ.

To be very precise we should now enrich our notations A, S, H, P etc. with an index p to indicate their dependence on p. However for any two p the corresponding sets of alcoves etc. are in natural bijection under a suitable homothety of E with center in (−ρ). Thus we hope it will not lead to confusion if we continue to write A, S, H, P etc. without an index. We will however introduce a notation for the identifications P → pX −ρ and call them λ_{p}. The action of ℤR ⊂ W on A extends to an action of X by translations, given by A ↦ A+pν for A ∈ A, ν ∈ X.

The choice of positive roots determines a choice of positive halfspace for each reflecting hyperplane H ∈ H. Namely for H = H_{α,m} with α ∈ R^{+}, m ∈ ℤ put H^{+} = {λ | ⟨λ + ρ, α^{∨}⟩ > pm}. This is the halfspace meeting all translates of the dominant Weyl chamber. Call the opposite open halfspace H^{−}. For a facet F one defines its upper closure

F̂ = \{λ ∈ \overline{F} | λ ∈ H ⇒ F ⊂ \overline{H^{−}} for all H ∈ H\}.

For every special point v ∈ P there is exactly one alcove A such that v lies in its upper closure, v ∈ Â. Call this alcove A = v^{−}.

Lemma 2.5.1. Let A^{−} ∈ A be the unique alcove containing (−ρ) in its upper closure and let A be any alcove contained in the “negative Weyl chamber”, i.e. A ⊂ H_{α,0}^{−} for all α ∈ R^{+}. Then there exists a sequence of alcoves A = A_{0}, A_{1}, …, A_{r} = A^{−} such that for suitable walls s_{1}, …, s_{r} ∈ S we have A_{i} = A_{i−1}s_{i} and A_{i} lies in the positive halfspace of the unique H ∈ H separating A_{i} and A_{i−1}.

Proof. Certainly we can find a point x ∈ A such that the line segment [x, −ρ] does not meet any facets except alcoves, walls, and the point −ρ. The alcoves met by this line segment obviously form the sequence needed.

2.6. The translation principle. ([Jan87], II.9.19.) The linkage principle together with the reciprocity principle tell us that [Q(λ) : L(μ)] ≠ 0 implies W ·_{p} λ = W ·_{p} μ. Thus any M ∈ C admits a unique direct sum decomposition M = ⨁_{Ω} M_{Ω} with Ω running over the (·_{p})-orbits in X, such that [M_{Ω} : L(λ)] ≠ 0 implies λ ∈ Ω. Namely M_{Ω} is the sum of all images of morphisms Q(λ) → M with λ ∈ Ω. We put C(Ω) = {M ∈ C | [M : L(λ)] ≠ 0 ⇒ λ ∈ Ω} and get even a direct sum decomposition of categories C = ⨁_{Ω} C(Ω). The corresponding projection and injection functors will be denoted by i_{Ω} : C(Ω) → C and pr_{Ω} : C → C(Ω), M ↦ M_{Ω}.

After these preparations we come to translation. Let Ω, Γ ⊂ X be two (·_{p})-orbits. The translation functor

T_{Ω}^{Γ} : C(Ω) → C(Γ)

is defined as the composition

T_{Ω}^{Γ} = pr_{Γ} ∘ (E_{Ω}^{Γ}⊗) ∘ i_{Ω};

Here E_{Ω}^{Γ} ∈ C is obtained as follows: Let λ and μ be the representatives of Ω and Γ in the closure Ā of a fixed alcove A ∈ A. Then E_{Ω}^{Γ} is the simple G-module with extremal weight μ − λ, i.e. with highest weight the dominant member of the Weyl group orbit (μ−λ). We may consider E_{Ω}^{Γ} as an object of C. It is clear that for M, N ∈ C the tensor product M ⊗_{k} N with its obvious 𝔤-action and X-grading again lies in C.

The translation functors are exact and commute with our duality. The right (and left) adjoint of T_{Ω}^{Γ} is just T_{Γ}^{Ω}. From this it follows that if we apply a translation functor to a projective object it stays projective. We say that Γ is on more walls than Ω and write Ω → Γ if and only if for each λ ∈ Ω there is some μ ∈ Γ which lies in the closure of the facet of λ. This μ then is unique and we denote it by λ_{Γ}.

Proposition 2.6.1. Suppose Γ is on more walls than Ω.

1.

T_{Ω}^{Γ}Z(λ) ≅ Z(λ_{Γ}) ∀λ ∈ Ω;

T_{Ω}^{Γ}L(λ) = \{ \begin{matrix}L(λ_{Γ}) & if λ_{Γ} is in the upper \\ & closure of the facet of λ; \\ 0 & else. \end{matrix}

2. T_{Γ}^{Ω}Z(μ) has a Z-filtration with multiplicities

(T_{Γ}^{Ω}Z(μ) : Z(λ)) = \{ \begin{matrix}1 & if λ ∈ Ω, λ_{Γ} = μ; \\ 0 & else. \end{matrix}

If Ω → Γ → Ω, then λ ↦ λ_{Γ} is in fact a bijection from Ω to Γ and T_{Ω}^{Γ} : C(Ω) → C(Γ) is an equivalence of categories.

A (·_{p})-orbit Ω is called regular if it does not meet any reflecting hyperplane. A regular orbit meets any alcove in exactly one point. The following are equivalent for l ∈ ℕ_{≥1} :

1. There exists a regular (·_{l})-orbit Ω ⊂ X.

2. The (·_{l})-orbit of 0 ∈ X is regular.

3. l is at least the Coxeter number h of our root system. This fact is the reason for the importance of the Coxeter number in our approach. Remark that the formulas for translation functors tell us how to determine all multiplicities [Z(λ) : L(μ)] with λ, μ ∈ X once we know them for all λ, μ in a given regular orbit.

2.7. Wall-crossing. Let now Ω ⊂ X be a regular orbit. For every s ∈ S we define a wall-crossing functor

Θ_{s} : C(Ω) → C(Ω)

as follows: Choose a (·_{p})-orbit Γ = Γ(s) ⊂ X which lies in s. This is always possible when there exist regular orbits, see [Jan87], II.6.3(1). Then put Θ_{s} = T_{Γ}^{Ω} ∘ T_{Ω}^{Γ}. One can show that this functor does not depend on the choice of Γ up to natural equivalence. However we won’t use this fact. From the properties of translation functors recalled above we deduce that Θ_{s} is exact, selfadjoint, commutes with the duality and maps projectives to projectives. We deduce in addition from Proposition 2.6.1 that for λ ∈ Ω the module Θ_{s}Z(λ) has a two-step filtration with subquotients Z(λ) and Z(λ′), where λ′ ∈ Ω is determined by the condition {λ, λ′} = {μ ∈ Ω | μ_{Γ} = λ_{Γ}}.

Let us reformulate this result in other notations. If p = chark ≥ h we associate to every alcove A ∈ A the Baby-Verma Z_{k}(A) = Z_{k}(λ), where λ ∈ X is given by {λ} = (W ·_{p} 0) ∩ A. We form furthermore its unique simple quotient L_{k}(A) = L_{k}(λ) and denote its projective cover by Q_{k}(A) = Q_{k}(λ). Then Θ_{s}Z_{k}(A) ≅ Θ_{s}Z_{k}(As) has a twostep filtration with subquotients Z_{k}(A), Z_{k}(As), and if A lies below the hyperplane H ∈ H which separates it from As there is a short exact sequence

Z_{k}(As) ↪ Θ_{s}Z_{k}(A) ↠ Z_{k}(A),

as is easily deduced from remark 2.3.2. In addition we claim that in this setup Θ_{s}L_{k}(As) = 0 and [Θ_{s}L_{k}(A) : L_{k}(As)] ≠ 0. Here the first statement follows directly from Proposition 2.6.1, whereas the second one can be deduced from

Hom_{C}(Z_{k}(As), Θ_{s}L_{k}(A)) = Hom_{C}(T_{Ω}^{Γ}Z_{k}(As), T_{Ω}^{Γ}L_{k}(A)) = k ≠ 0.

We continue to assume p ≥ h. We then write T^{out} for the translation functor from any orbit to the regular orbit W ·_{p} 0. For any special point v ∈ P let us abbreviate Z_{k}(λ_{p}(v)) = Z_{k}(v).

Lemma 2.7.1. We have Q_{k}(v^{−}) ≅ T^{out}Z_{k}(v). In particular we find (Q_{k}(v^{−}) : Z_{k}(A)) = 1 if v ∈ Ā, zero else.

Proof. The second statement follows from the first (using Proposition 2.6.1 (ii)). Let Ω, Γ be the (·_{p})-orbits of 0, v. By Lemma 2.4.2 the Baby-Verma Z_{k}(v) is projective, hence so is T^{out}Z_{k}(v) = T_{Γ}^{Ω}Z_{k}(λ_{p}(v)). From Proposition 2.6.1 (i) we deduce T_{Ω}^{Γ}L_{k}(A) ≅ L_{k}(λ_{p}(v)) if v ∈ Â, i.e. A = v^{−}, and zero else. We deduce

Hom_{C}(T_{Γ}^{Ω}Z_{k}(v), L_{k}(A)) = Hom_{C}(Z_{k}(v), T_{Ω}^{Γ}L_{k}(A)) = \{ \begin{matrix}k & if A = v^{−}; \\ 0 & else. \end{matrix}

Thus indeed T^{out}Z_{k}(v) ≅ Q_{k}(v^{−}).

2.8. The quantum analogue. ([AJS94], 2–7 with A = k.) The theory exposed in the preceding sections has a complete analogue in characteristic zero. Namely suppose given a root system R; a system of a simple roots Δ ⊂ R; an integer p > 1 prime to all entries of the Cartan matrix, hence odd; and a primitive p-th root of unity ζ. To these data Lusztig [Lus90b], 8.2 associates the “restricted quantum group”, a finite dimensional ℤR-graded algebra u over k = ℚ(ζ).

Its degree-zero part contains the group ring of ℤR/2pℤR and for α ∈ R we denote by K_{α} ∈ u the corresponding element. In this case we put

C = C_{k} = \{M ∈ u-mod_{X} | \begin{matrix}M is finitely generated and \\ K_{α}m = ζ^{d_{α}⟨ν,α^{∨}⟩}m \\ ∀ α ∈ Δ, ν ∈ X, m ∈ M_{ν} \end{matrix} \}

where as usual {d_{α}}_{α∈Δ} are the unique integers in {1, 2, 3} such that the matrix (d_{α}⟨β, α^{∨}⟩)_{α,β∈Δ} is symmetric and each irreducible component of R contains an α ∈ Δ with d_{α} = 1. The theory explained in the previous sections has a complete analogue in in this setting, just some proofs and constructions become opaque. For example one can construct for any λ ∈ X an object Z(λ) = Z_{k}(λ) ∈ C_{k} such that

1. dim Z(λ)_{λ} = 1

2. Z(λ)_{ν} ≠ 0 ⇒ ν ∈ λ − ℕR^{+} and

3. if M ∈ C is such that M_{ν} = 0 whenever ν ∈ λ + ℕR^{+} and ν ≠ λ, then the map Hom_{C}(Z(λ), M) → M_{λ}, f ↦ f(v_{λ}) is a bijection, for any basis v_{λ} of Z(λ)_{λ}.

Clearly the pair (Z(λ), v_{λ}) is characterized by these conditions up to unique isomorphism. Again the Z(λ) have unique simple quotients L(λ), these L(λ) are just the simples of C, the linkage principle and reciprocity principle hold, one can construct a duality and translation functors with the same properties: The only change is that we have to consider the (·_{p})-action of W on X, where p is the integer involved in the construction of u. For p ≥ h we again associate to every alcove A the objects Z_{k}(A), L_{k}(A) and Q_{k}(A) to be just Z_{k}(λ), L_{k}(λ), Q_{k}(λ) with λ the unique representative of W ·_{p} 0 in A.

3. The main results

3.1. Independence of the characteristic. Let us start to explain independence of the characteristic in the simplest context. If A is a finite dimensional (associative unitary) algebra over a field, it may decompose into a product of subalgebras, A = ∏_{B} B. There is a unique maximal such decomposition and the corresponding subalgebras B are called the blocks of A. Any indecomposable A-module M is annihilated by all blocks except one: the block of M. In a representation theoretic context the block of the trivial onedimensional representation is called the “principal block”.

Let us also recall the notion of Morita equivalence. Let k be a field. Two finite dimensional k-algebras are said to be Morita-equivalent if and only if their respective categories of finite dimensional modules are equivalent k-categories. For example the matrix algebras of various sizes are all Morita equivalent to the ground field.

Fix now a root system R with Coxeter number h. Put ℤ′ = ℤ[n^{−1} | 1 < n < h]. The following is [AJS94], 16.18.

Theorem 3.1.1. There exists a ℤ′-algebra E, free of finite rank as a ℤ′-module, such that

1. for any algebraically closed field k of characteristic chark > h the k-algebra E ⊗_{ℤ} k is Morita equivalent to the principal block of the restricted enveloping algebra U^{res}(𝔤) over k corresponding to R, 2. for any integer p > h prime to all entries of the Cartan matrix

of R and any primitive p-th root of unity ζ ∈ ℂ, the ℚ(ζ)-algebra E ⊗_{ℤ} ℚ(ζ) is Morita equivalent to the principal block of the restricted quantum group u over ℚ(ζ) corresponding to R.

This is not easy to prove, so let us start with a simpler task in form of a lemma. Although this lemma is not needed for the logic of our argument, we hope that its proof will clarify our strategy.

Lemma 3.1.2. We can associate to each algebraically closed field k of characteristic chark > h a k-algebra E_{k} which is Morita equivalent to the principal block of the corresponding U^{res}(𝔤), in such a way that dim_{k} E_{k} is independent of k.

Proof. Consider our category C = C_{k} and the (·_{p})-orbit Ω = Ω_{k} = W ·_{p} 0 ⊂ X. Since we assumed p > h it is regular (p ≥ h would be enough). Consider Q_{k} = Q_{k}(A^{−}) = T^{out}Z_{k}(−ρ). For any sequence σ : [1, n] → S consider the iterated wall crossing functor Θ_{σ} = Θ_{σ(1)} … Θ_{σ(n)}. Let |σ| denote the length n of σ. We claim that for sufficiently big N the k-algebras E_{k} = E_{k,N} = End_{𝔤}(⨁_{|σ|≤N} Θ_{σ}Q_{k})^{opp} meet our requirements. To establish this we need to prove

1. There exists N ≫ 0 such that for all k with chark > h the kalgebra E_{k,N} is Morita equivalent to the principal block of U^{res}(𝔤).

2. The dimension dim_{k} E_{k,N} is independent of k. We start with (i). It is known that the simple modules of the principal block of U^{res}(𝔤) are the L(\overline{x · 0}) for x ∈ W. In particular, any M ∈ C(Ω) belongs to the principal block of U^{res}(𝔤) when we forget the X-grading on M. It is also clear that any projective object in C is projective as an U^{res}(𝔤)-module. Since Q_{k} = Q_{k}(A^{−}) is projective so are all Θ_{σ}Q_{k}. Thus it only remains to show that for big N our sum ⨁_{|σ|≤N} Θ_{σ}Q_{k} is a generator of the principal block of U^{res}(𝔤), i.e. that for each x ∈ W the simple object L(\overline{x · 0}) is a quotient of some Θ_{σ}Q_{k}. We now apply Lemma 2.5.1. Choose an alcove A in the negative Weyl chamber such that x·0+pℤR meets A. Choose a sequence A = A_{0}, A_{1}, …, A_{r} = A^{−} of alcoves as in the lemma, with A_{i} = A_{i−1}s_{i} for some s_{i} ∈ S. Section 2.7 tells us that

[Θ_{s_{r}} … Θ_{s_{1}}L_{k}(A) : L_{k}(A^{−})] ≠ 0,

thus

0 ≠ Hom_{C}(Q_{k}, Θ_{s_{r}} … Θ_{s_{1}}L_{k}(A)) = Hom_{C}(Θ_{s_{1}} … Θ_{s_{r}}Q_{k}, L_{k}(A))

and thus indeed Hom_{𝔤}(Θ_{σ}Q_{k}, L(\overline{x · 0})) ≠ 0 for a suitable sequence σ.

Next comes (ii). First we claim that for any M, N ∈ C(Ω) we have

Hom_{𝔤}(M, N) = ⨁_{ν∈ℤR}Hom_{C}(M, N[pν]).

From the breakup U^{res}(𝔤)-mod_{X}^{fg} = ⨁ C[μ] with μ running over X/pX it is clear that we have Hom_{𝔤}(M, N) = ⨁_{ν∈X} Hom_{C}(M, N[pν]). For ν ∈ X one verifies easily that Z(λ)[pν] ≅ Z(λ + pν), L(λ)[pν] ≅ L(λ + pν) and thus C(Ω)[pν] = C(Ω + pν). Since we required p > h we know that Ω + pν = Ω implies ν ∈ ℤR. So indeed Hom_{C}(M, N[pν]) = 0 for ν ∉ ℤR.

It remains to be shown that for any two sequences σ, τ and ν ∈ ℤR the dimension of Hom_{C}(Θ_{σ}Q_{k}, Θ_{τ}Q_{k}[pν]) is independent of k. But remark that for any two projectives P, Q ∈ C we have dim Hom_{C}(P, Q) = ∑_{λ}(Q : Z(λ))(P : Z(λ)), as can be seen from the reciprocity principle: Namely it is enough to show this equality for indecomposable P, e.g. P = Q(ν), and then we calculate

dim Hom_{C}(Q(ν), Q) = [Q : L(ν)]
= ∑_{λ}(Q : Z(λ))[Z(λ) : L(ν)]
= ∑_{λ}(Q : Z(λ))(Q(ν) : Z(λ)).

(Another way would be to use selfduality P^{τ} ≅ P along with Lemma 2.3.1.) But it is clear from the properties of translation functors that the Verma-flag multiplicities (Θ_{σ}Q_{k} : Z_{k}(A)) depend only on σ and A, not on p or k. So indeed dim_{k} E_{k,N} is independent of k.

We are now in position to explain precisely what independence of the characteristic we can prove. For our root system R with Coxeter number h let us introduce the class of fields Bon(R) consisting of

1. all algebraically closed fields k of characteristic p = chark ≥ h and

2. all fields k = ℚ(ζ) where ζ is a primitive p-th root of unity in ℂ such that p ≥ h and p is prime to all entries of the Cartan matrix of R.

To any k ∈ Bon(R) is associated a number p = p(k). Let I be a finite set consisting of pairs (σ, v) where σ is a finite sequence in S and v ∈ P is a special point. For any k ∈ Bon(R) we can construct the k^{I}-algebra

B_{k} = End_{C}(⨁ Θ_{σ}Q_{k}(v^{−}))

where we sum over all (σ, v) ∈ I.

Theorem 3.1.3. There exists a (ℤ′)^{I}-algebra B, free of finite rank as a ℤ′-module, such that B_{k} ≅ B ⊗_{ℤ} k for all k ∈ Bon(R).

We also give a more precise variant, so that the reader may deduce Theorem 3.1.1. Define for M ∈ C the ℤR-graded ring End_{C}^{♯}M = ⨁_{ν∈ℤR} Hom_{C}(M, M[pν]). Consider the ℤR-graded k^{I}-algebra

E_{k} = End_{C}^{♯}(⨁ Θ_{σ}Q_{k}(v^{−})).

Theorem 3.1.4. There exists a ℤR-graded (ℤ′)^{I}-algebra E, free of finite rank as a ℤ′-module, such that E_{k} ≅ E ⊗_{ℤ} k for all k ∈ Bon(R).

The remainder of this article splits into two independent parts. In the following subsection we will explore the consequences of our theorems. After that we will start a new section to explain the proof.

3.2. Composition series of Verma modules. The following theorem is what we were really after.

Theorem 3.2.1. For any root system R there exists a bound b(R) such that over all k ∈ Bon(R) with chark = 0 or chark ≥ b(R) the multiplicities [Z_{k}(B) : L_{k}(A)] are constant in k and depend only on the alcoves A, B ∈ A.

The point is that these multiplicities are known for chark = 0 in the simply laced case. Using the theorem we can get them for chark ≥ b(R) as well.

Proof. Recall that Z_{k}(λ)[pν] = Z_{k}(λ + pν), L_{k}(λ)[pν] = L_{k}(λ + pν) for all ν ∈ X. Thus it is sufficient to compute [Z_{k}(λ) : L_{k}(μ)] for all μ in a system of representatives of X/pX. Recall furthermore that by the translation principle if we know the multiplicities [Z_{k}(λ) : L_{k}(μ)] for some regular μ, we know them for all μ in the closure of its alcove. Thus it suffices to compute the [Z_{k}(B) : L_{k}(A)] with A running over a system of representatives of the X-orbits in A.

As shown in [AJS94], 16.20 it is possible to find alcoves A_{1}, …, A_{r} ∈ A and special points v_{1} …, v_{r} ∈ P and sequences σ_{1}, …, σ_{r} in S such that

Θ_{σ_{i}}Q_{k}(v_{i}^{−}) ≅ Q_{k}(A_{i}) ⊕ ⨁_{j<i} Q_{k}(A_{j})^{m_{k}(i,j)}

for suitable m_{k}(i, j), and such that each X-orbit in A contains at least one of the A_{i}. Now we claim that it suffices to show that the m_{k}(i, j) are independent of k, for chark = 0 or chark ≫ 0. Indeed, since by subsection 2.7 the Verma flag multiplicities on the left hand side are independent of k, this would mean that by induction on i the Verma flag multipicities (Q_{k}(A_{i}) : Z_{k}(B)) are independent of k as well, for chark = 0 or chark ≫ 0, and the theorem would follow via the reciprocity principle.

So why are the m_{k}(i, j) independent of k, for chark = 0 or chark ≫ 0? We will deduce this from Theorem 3.1.3 and have to recall some generalities from ring theory. For a finite dimensional algebra A over a field k the following statements are equivalent:

1. The only idempotents are 0, 1 and 0 ≠ 1.

2. A is indecomposable when considered as a right A-module.

3. A is indecomposable when considered as a left A-module.

4. A is local, i.e. all noninvertible elements form an ideal (and 1 ≠ 0). As the basic example, note that a finite length object M of some abelian category is indecomposable if and only if its endomorphism ring is local [Ben91], 1.4.5. Let us call A “absolutely local” if it stays local under any field extension k′/k.

Suppose now 𝔬 ⊃ 𝔪 is a discrete valuation ring with maximal ideal 𝔪 and A is an 𝔬-algebra, free of finite rank as an 𝔬-module.

Lemma 3.2.2. If A ⊗_{𝔬} Quot𝔬 is absolutely local, then A ⊗_{𝔬} (𝔬/𝔪) is local.

Proof. This is a direct consequence of “lifting of idempotents”. Namely consider the 𝔪-adic completion 𝔬̂ of 𝔬. It is known ([Ben91], 1.9.4) that each idempotent e of A ⊗_{𝔬} (𝔬/𝔪) lifts to an idempotent of A ⊗_{𝔬} 𝔬̂ and thus leads to an idempotent in A ⊗_{𝔬}Quot𝔬̂. This has to be zero or one, by assumption. Thus e itself is zero or one.

An idempotent e of a ring is called “primitive” if it is nonzero and does not admit a decomposition e = e_{1} + e_{2} into two nonzero idempotents that are orthogonal, i.e. satisfy e_{1}e_{2} = e_{2}e_{1} = 0. As the basic example, if M is a finite length object in some abelian category, then e ∈ EndM is a primitive idempotent if and only if it is a projection onto an indecomposable summand of M. If our ring is an algebra over a field, call e “absolutely primitive” if it stays primitive under any field extension. Let now 𝔬 ⊃ 𝔪 and A be as in the previous lemma.

Lemma 3.2.3. Let e ∈ A be an idempotent. If the image of e in A ⊗_{𝔬} Quot𝔬 is absolutely primitive, then the image of e in A ⊗_{𝔬} (𝔬/𝔪) is primitive.

Proof. By our definitions, an idempotent e of a finite dimensional algebra B over a field is (absolutely) primitive if and only if the algebra eBe is (absolutely) local. The lemma now follows from the previous one.

To finish the preparations, let M be any object of an additive category with a decomposition M = ⨁_{i=1}^{n} N_{i}. Let π_{i} ∈ EndM be the corresponding projections. Then we have N_{i} ≅ N_{j} if and only if π_{i} is conjugate to π_{j} by a unit in EndM.

Now we can explain why the m_{k}(i, j) do not depend on k for chark = 0 or chark ≫ 0. Namely consider the k-algebra k^{n} = k × … × k. We write 1_{i} for the idempotent (0, …, 1, …, 0) with a one at the i-th place. Consider the k^{n}-algebra B_{k} = End_{C}(⨁_{i=1}^{n} Θ_{σ_{i}}Q_{k}(v_{i}^{−})), where 1_{i} ∈ k^{n} acts as projection onto the i-th factor. The m_{k}(i, j) can be read off from the k^{n}-algebra B_{k} as follows: Write each 1_{i} ∈ B_{k} as a sum 1_{i} = e_{i}^{0} + … + e_{i}^{r_{i}} of pairwise orthogonal primitive idempotents. Exactly one of the e_{i}^{s}, say e_{i}^{0}, won’t be conjugate to any e_{j}^{t} with j < i. (One should think of e_{i}^{0} as the projection of Θ_{σ_{i}}Q_{k}(v_{i}^{−}) onto Q_{k}(A_{i}).) Now m_{k}(i, j) is the number of idempotents among the e_{i}^{s} which are conjugate to e_{j}^{0}.

By Theorem 3.1.3 we have a (ℤ′)^{n}-algebra B, free of finite rank as a ℤ′-module, such that B ⊗_{ℤ} k ≅ B_{k} for all our k. Choose now for some k = ℚ(ζ) a system of e_{i}^{s}. For suitable b ∈ ℤ, all e_{i}^{s} and all conjugacy relations among them come from elements ẽ_{i}^{s} ∈ B ⊗_{ℤ}ℤ[ζ, b^{−1}] and conjugacy relations in this ring. The e_{i}^{s} ∈ B_{ℚ(ζ)} are absolutely primitive, since the endomorphisms of simple objects of C_{ℚ(ζ)} are reduced to scalars. We can then deduce from Lemma 3.2.3 that the ẽ_{i}^{s} give primitive idempotents ē_{i}^{s} in B_{k} whenever b is invertible in k. Clearly the ē_{i}^{s} are conjugate if the e_{i}^{s} are. But there can be no additional conjugacy relations among the ē_{i}^{s}, since the number n of conjugacy classes of primitive idempotents in B_{k} is independent of k. From the description of the m_{k}(i, j) in terms of the structure of the k^{n}-algebra B_{k} given above it is now clear that m_{k}(i, j) is independent of k over all k ∈ Bon(R) such that b is invertible in k.

4. Combinatorics

4.1. The deformed representation categories. At the basis of our whole work is the idea to “deform” the category C, thus to think of the Z(λ) as embedded in a whole family of objects Z(λ + ε) for “small” ε in 𝔥^{∗}. These Z(λ + ε) however will no longer be restricted 𝔤-modules resp. u-modules. In fact, the whole category C can be viewed as the special fibre at ε = 0 of a flat family C_{ε}. For generic ε this category is semisimple with simple objects the Z(λ + ε), so everything is very easy. For subgeneric ε (fixed by just one reflection of W) the category C_{ε} is still tractable and looks like an 𝔰𝔩_{2}-category. Since we have a flat family, once we know it for all ε on at most one reflecting hyperplane, we know it everywhere and can recover the most singular fibre at the point ε = 0, which is just our category C.

We now put this idea into formulas. Let us treat the case of positive characteristic first. Here instead of the restricted enveloping algebra U^{res}(𝔤) we consider the bigger algebra U = U(𝔤)/I where I ⊂ U(𝔤) is the twosided ideal generated by all D^{[p]} − D^{p} for D ∈ 𝔤 root vectors, i.e. D ∈ 𝔤_{α} for some α ∈ R. The algebra U inherits an X-grading from U(𝔤). Let S(𝔥) = S be the symmetric algebra over 𝔥. The injection S ↪ U(𝔤) gives us an injection S ↪ U, and U is a free S-module of rank p^{dim 𝔤−dim 𝔥}. For later use remark also that the map 𝔟 ↠ 𝔥 leads to a morphism U(𝔟) → S = U(𝔥).

Let A be any commutative noetherian S-algebra with structural homomorphism π : S → A. We consider A as an X-graded algebra concentrated in degree zero and define the deformation category C_{A} to consist of all finitely generated X-graded U-A-bimodules M such that

Hm = m(π(H) + ν̄(H))

for all H ∈ 𝔥, ν ∈ X, m ∈ M_{ν}. As before we denote here by ν̄ ∈ 𝔥^{∗} the differential of ν ∈ X. It is clear that for A = k with π the augmentation we get back our category C_{k}. In C_{A} we define for any λ ∈ X the “deformed Baby-Verma with highest weight λ” by the formula

Z_{A}(λ) = U ⊗_{U(𝔟)} (k_{λ} ⊗_{k} A).

Here U(𝔟) acts on k_{λ} as explained before, on A via the composition U(𝔟) → S → A, and (k_{λ} ⊗_{k} A) is understood to be the tensor product representation. The right action of A is understood to just affect the third factor of this tensor product, and the X-grading is determined by giving (k_{λ} ⊗_{k} A) the degree λ.

The quantum case is similar and we refer to [AJS94], 1.3 for the details. Let us just note that one constructs in analogy with the case of positive characteristic a ℤR-graded ℚ(ζ)-algebra U along with a homogeneous surjection U ↠ u. The degree-zero part of U contains the group ring of ℤR and we denote by K_{α} ∈ U the element corrresponding to α ∈ R. Consider the ring of Laurent polynomials S = ℚ(ζ)[K_{α}, K_{α}^{−1}] ⊂ U. One might view S as the ring of regular functions on a torus. For any commutative noetherian S-algebra A with structural morphism π : S → A we define C_{A} to be the category of all finitely generated X-graded U-A-bimodules M such that K_{α}m = m(π(K_{α})ζ^{d_{α}⟨ν,α^{∨}⟩}) for all α ∈ R, ν ∈ X, m ∈ M_{ν}. For A = ℚ(ζ) = k with π the “evaluation at the identity”, i.e. π(K_{α}) = 1 for all α, we just get back our old C_{k}.

Homomorphisms in C_{A} will just be written Hom_{C}. If A′ is a commutative noetherian A-algebra, we have a base change functor ⊗_{A}A′ :

C_{A} → C_{A′} that maps Z_{A}(λ) to Z_{A′}(λ). The following proposition collects some basic properties of our deformation categories. For the proof we refer to [AJS94], 2.13 and section 3.

Proposition 4.1.1. Let A, A′ be as above.

1. There are enough projectives in C_{A}.

2. If P, Q ∈ C_{A} are projective, then Hom_{C}(P, Q) is a projective Amodule.

3. If P ∈ C_{A} is projective, so is P ⊗_{A} A′ ∈ C_{A′}.

4. If P, Q ∈ C_{A} are projective, the natural map

Hom_{C}(P, Q) ⊗_{A} A′ → Hom_{C}(P ⊗_{A} A′, Q ⊗_{A} A′)

is an isomorphism.

Any A determines a partition of X into the so called A-blocks (see [AJS94], section 6). We define this to be the finest possible partition such that λ and μ belong to the same A-block if Hom_{C}(Z_{A}(λ), Z_{A}(μ)) or Ext_{C}^{1}(Z_{A}(λ), Z_{A}(μ)) do not vanish. We can then prove that we have a decomposition of categories C_{A} = ⨁ C_{A}(Ω) with Ω running over the A-blocks, where C_{A}(Ω) denotes the category of all quotients of objects with a Z_{A}-filtration such that all subquotients Z_{A}(λ) have parameter λ ∈ Ω. We still use the notation C_{A}(Ω) if Ω is a union of A-blocks. If A′ is an A-algebra, each A-block is a union of A′-blocks. Remark that the partition of X into k-blocks refines the partition into (·_{p})-orbits. For example each special point is a k-block by itself.

Let us consider now the special case where A = Ŝ is the completion of S at the kernel of our map S → k from above. We prove that the Ŝ-blocks coincide with the k-blocks. So for any two (·_{p})-orbits Ω, Γ ⊂ X we can define as in subsection 2.6 (see [AJS94], section 7) the translation functor T_{Ω}^{Γ} : CŜ(Ω) → CŜ(Γ), as the composition T_{Ω}^{Γ} = pr_{Γ} ∘ (E_{Ω}^{Γ}⊗) ∘ i_{Ω} (where we should think a bit why for E ∈ C, M ∈ C_{A} the tensor product E ⊗_{k} M is in a natural way an object of C_{A}) and the functorial diagrams

CŜ(Ω) ⟶^{T_{Ω}^{Γ}} CŜ(Γ)
↓^{⊗Ŝk}  ↓^{⊗Ŝk}
C_{k}(Ω) ⟶^{T_{Ω}^{Γ}} C_{k}(Γ)

commute. Again the left (and right) adjoint of T_{Ω}^{Γ} is T_{Γ}^{Ω}. Again our translation functors are exact and (by having an exact adjoint functor) transform projectives to projectives. The formulas from Proposition 2.6.1 for translation of Z-objects generalize to our deformed situation without essential changes. Again we can define the wall-crossings Θ_{s} : CŜ(Ω) → CŜ(Ω) for any regular (·_{p})-orbit Ω ⊂ X. For any special point v ∈ pX − ρ the deformed Verma ZŜ(v) = ZŜ(λ_{p}(v)) is projective in CŜ.

Assume now p ≥ h and recall our k^{I}-algebra

B_{k} = End_{C}(⨁ Θ_{σ}Q_{k}(v^{−})) = End_{C}(⨁ Θ_{σ}T^{out}Z_{k}(v))

from the end of subsection 3.1. By Proposition 4.1.1 (iv) we may view it as a special fibre of the Ŝ^{I}-algebra BŜ = End_{C}(⨁ Θ_{σ}T^{out}ZŜ(v)), in other words B_{k} ≅ BŜ ⊗Ŝ k. Remark that by Proposition 4.1.1 (ii) the algebra BŜ is free over Ŝ, so we really deformed B_{k} into a flat family.

Now we have to work with variable k, so we write Ŝ = Ŝ(k) to indicate its dependence on k. Let S(ℤ) denote the symmetric algebra over ℤR and put S(ℤ′) = S(ℤ) ⊗_{ℤ} ℤ′. For α ∈ R denote by h_{α} the corresponding element of S(ℤ′). We define morphisms S(ℤ′) → Ŝ(k) for all k ∈ Bon(R): In case chark > 0 map h_{α} to d_{α}H_{α}, where H_{α} ∈ 𝔥 is the image of the coroot α^{∨}. In case chark = 0 we map h_{α} to log K_{α} (viewed as power series in (K_{α} − 1), thus as an element of Ŝ(k)). We deduce Theorem 3.1.3 from the following more detailed

Theorem 4.1.2. There exists a S(ℤ′)^{I}-algebra B_{S(ℤ′)}, free of finite rank as an S(ℤ′)-module, such that

B_{Ŝ(k)} ≅ B_{S(ℤ′)} ⊗_{S(ℤ′)} Ŝ(k)

for every k ∈ Bon(R).

From this we obtain Theorem 3.1.3 just putting B = B_{S(ℤ′)} ⊗_{S(ℤ′)} ℤ′. So we should explain how to construct B_{S(ℤ′)}. The point is to give some combinatorial description for the algebras B_{Ŝ(k)}. This description will make sense also over ℤ′ and construct for us the algebra B_{S(ℤ′)}.

4.2. The functor into combinatorics. Let us fix again k and write Ŝ(k) = Ŝ. The images of the h_{α} in Ŝ will be denoted h_{α} as well. To describe the algebras BŜ we have to describe homomorphisms between projective objects P, Q ∈ CŜ. This task we attack as follows. For any β ∈ R^{+} put Ŝ^{β} = Ŝ[h_{α}^{−1} | α ∈ R^{+}, α ≠ β] and define Ŝ^{∅} = Ŝ[h_{α}^{−1} | α ∈ R^{+}]. If our characteristic is prime to all entries of the Cartan matrix, we have Ŝ = ⋂_{β∈R^{+}} Ŝ^{β} inside Ŝ^{∅}. For any Ŝ-module M put M^{β} = M ⊗Ŝ Ŝ^{β}, M^{∅} = M ⊗Ŝ Ŝ^{∅}. Using Proposition 4.1.1 we deduce for any two projectives P, Q ∈ CŜ

Hom_{C}(P, Q) = ⋂_{β∈R^{+}} Hom_{C}(P^{β}, Q^{β}) ⊂ Hom_{C}(P^{∅}, Q^{∅}).

Let us abbreviate CŜ^{β} = C^{β} and ZŜ^{β}(λ) = Z^{β}(λ). Remark that SpecŜ^{β} is just SpecŜ with all walls except the β-wall deleted, so it is not surprising that C^{β} behaves like a category of representations of 𝔰𝔩_{2}. More precisely, for every λ ∈ X define β ↑ λ = s_{H}λ where H is the lowest reflecting hyperplane orthogonal to β such that λ lies below or on H, i.e. H = H_{β,m} where m is minimal with ⟨λ + ρ, β^{∨}⟩ ≤ pm. Then λ ↦ β ↑ λ is a bijection. We denote its inverse by λ ↦ β ↓ λ. In [AJS94], section 8 we prove:

Proposition 4.2.1. 1. If β ↑ λ = λ then Z^{β}(λ) is projective in C^{β}.

2. If β ↑ λ ≠ λ then Ext_{C}^{1}(Z^{β}(λ), Z^{β}(β ↑ λ)) is isomorphic to Ŝ^{β}/h_{β}Ŝ^{β}, and if we represent an extension by a short exact sequence

Z^{β}(β ↑ λ) ↪ Q ↠ Z^{β}(λ),

then Q is projective in C^{β} if and only if our extension generates Ext^{1} as an Ŝ^{β}-module.

3. The projectives Z^{β}(λ) and Q from (i) and (ii) generate C^{β}.

4. The Ŝ^{β}-blocks are just the smallest sets stable under β ↑ and β ↓. In particular they all lie entirely inside a line with direction β.

The category CŜ^{∅} = C^{∅} is even simpler. Let us abbreviate ZŜ^{∅}(λ) = Z^{∅}(λ) ∈ C^{∅}. As proved in [AJS94], the remark closing 6.13 we have

Proposition 4.2.2. The Z^{∅}(λ) are all projective and generate C^{∅}. The Ŝ^{∅}-blocks are just points.

Now we want to glue together all the C^{β} to obtain a description of C. To put this intuition into formulas we introduce for any (·_{p})-orbit Ω the “combinatorial category” K(Ω, Ŝ). An object M ∈ K(Ω, Ŝ) consists by definition of the following data:

1. A collection M(λ), λ ∈ Ω of finitely generated Ŝ^{∅}-modules such that M(λ) = 0 for all but finitely many λ ∈ Ω.

2. For each β ∈ R^{+}, λ ∈ Ω a finitely generated Ŝ^{β}-submodule M(λ, β) of M(λ) ⊕ M(β ↑ λ) resp. M(λ) if β ↑ λ ≠ λ resp. β ↑ λ = λ.

A morphism φ : M → N is a collection of Ŝ^{∅}-linear maps φ_{λ} : M(λ) → N (λ) such that φ_{λ} ⊕φ_{β↑λ} resp. φ_{λ} maps M(λ, β) to N (λ, β) if β ↑ λ ≠ λ resp. β ↑ λ = λ. Morphisms in K(Ω, Ŝ) will just be written Hom_{K}.

Next we construct a functor

𝕍_{Ω} = 𝕍 : CŜ(Ω) ⟶ K(Ω, Ŝ).

It will depend on the choice of a collection e^{β}(λ) of generators of the Ŝ^{β}-modules Ext_{C}^{1}(Z^{β}(λ), Z^{β}(β ↑ λ)), one for each pair (λ, β) ∈ Ω × R^{+} such that β ↑ λ ≠ λ. Namely for M ∈ CŜ(Ω) put (𝕍M)(λ) = Hom_{C}(Z^{∅}(λ), M^{∅}). If β ↑ λ = λ put (𝕍M)(λ, β) = Hom_{C}(Z^{β}(λ), M^{β}). If finally β ↑ λ ≠ λ we choose a representative Z^{β}(β ↑ λ) ↪ Q^{β}(λ) ↠ Z^{β}(λ) of e^{β}(λ). If we extend scalars to Ŝ^{∅}, then by Proposition 4.2.2 this sequence splits canonically to determine an isomorphism

Q^{β}(λ)^{∅} = Z^{∅}(β ↑ λ) ⊕ Z^{∅}(λ).

Now let (𝕍M)(λ, β) be the image of Hom_{C}(Q^{β}(λ), M^{β}) in

Hom_{C}(Q^{β}(λ)^{∅}, M^{∅}) = (𝕍M)(λ) ⊕ (𝕍M)(β ↑ λ).

As a simple example, define for v ∈ P the object

ZŜ(v) ∈ K(W ·_{p} λ_{p}(v), Ŝ)

by the formulas

(ZŜ(v))(μ) = \{ \begin{matrix}Ŝ^{∅} & if μ = λ_{p}(v); \\ 0 & else, \end{matrix}

and

(ZŜ(v))(μ, β) = \{ \begin{matrix}Ŝ^{β} & if μ = λ_{p}(v); \\ 0 & else. \end{matrix}

Then we have 𝕍ZŜ(v) ≅ ZŜ(v).

The following Theorem is as easy to prove as it is central to our approach.

Theorem 4.2.3. Suppose the characteristic is prime to all entries of the Cartan matrix. Let Ω be a (·_{p})-orbit and suppose M, N ∈ CŜ(Ω) are projective as Ŝ-modules. Then the functor 𝕍 induces an isomorphism

Hom_{C}(M, N) →^{∼} Hom_{K}(𝕍M, 𝕍N).

Proof. Start with any λ ∈ Ω. By Proposition 4.2.2 it is a Ŝ^{∅}-block by itself, so we can decompose C^{∅}(Ω) = ⨁_{λ∈Ω} C^{∅}(λ). Furthermore Z^{∅}(λ) is a projective generator of C^{∅}(λ) and End_{C}Z^{∅}(λ) = Ŝ^{∅}, so Hom_{C}(Z^{∅}(λ), ) : C^{∅}(λ) → Ŝ^{∅}-mod is fully faithful. This means that we get a bijection

Hom_{C}(M^{∅}, N^{∅}) →^{∼} ⨁_{λ∈Ω} HomŜ^{∅}(𝕍M(λ), 𝕍N(λ))

and the pretended isomorphism from the theorem already has to be an injection. Now an element (φ_{λ})_{λ∈Ω} on the right hand side of our bijection maps (𝕍M)(λ, β) to (𝕍N)(λ, β) if and only if composition with the corresponding morphism φ : M^{∅} → N^{∅} on the left hand side maps Hom_{C}(Q^{β}(λ), M^{β}) ⊂ Hom_{C}(Q^{β}(λ)^{∅}, M^{∅}) to Hom_{C}(Q^{β}(λ), N^{β}) ⊂ Hom_{C}(Q^{β}(λ)^{∅}, N^{∅}), where we put Q^{β}(λ) = Z^{β}(λ) in case β ↑ λ = λ.

Since the Q^{β}(λ) generate C^{β}(Ω), we deduce that φ(M^{β}) ⊂ N^{β} for all β and hence φ(M) ⊂ N.

4.3. Combinatorics and Translations. Using the theorem, we can now rewrite the algebras at the center of our interest as

BŜ = End_{C}(⨁ Θ_{σ}T^{out}ZŜ(v))
= End_{K}(⨁ 𝕍Θ_{σ}T^{out}ZŜ(v)).

To describe them in a combinatorial way, we should describe 𝕍_{Γ}T_{Ω}^{Γ}M in terms of 𝕍_{Ω}M. Certainly this will depend on the choices of extensions used to define 𝕍_{Ω} and 𝕍_{Γ}. For simplicity we only treat translation from or to a regular orbit.

Let us start defining the combinatorial translations. Let Ω, Γ be two (·_{p})-orbits and suppose Ω is regular. For any H ∈ H let α(H) ∈ R^{+} be the orthogonal root, i.e. the root α ∈ R^{+} such that H = H_{α,m} for some m. For β ∈ R^{+} define H(β) = {H ∈ H | s_{β}α(H) ∉ R^{+}}. For any λ ∈ Ω, β ∈ R^{+} define a_{λ}^{β} ∈ Ŝ^{∅} by the formula

a_{λ}^{β} = ∏_{\begin{matrix}H∈H(β) \\ λ_{Γ}∈H, λ∈H^{+} \end{matrix}} h_{−α(H)} ∏_{\begin{matrix}H∈H(β) \\ λ_{Γ}∈H, λ∈H^{−} \end{matrix}} h_{α(H)}^{−1}

and put b_{λ}^{β} = (a_{λ}^{β})^{−1}. Then define the “combinatorial translation to the walls”

T = T_{Ω}^{Γ} : K(Ω, Ŝ) → K(Γ, Ŝ)

by the prescriptions

(T M)(μ) = ⨁_{λ_{Γ}=μ} M(λ)
(T M)(μ, β) = \{ \begin{matrix}⨁_{λ_{Γ}=μ=(β↑λ)_{Γ}}(b_{λ}^{β} ⊕ id)M(λ, β) & if β ↑ μ = μ; \\ ⨁_{λ_{Γ}=μ}(b_{λ}^{β} ⊕ id)M(λ, β) & if β ↑ μ ≠ μ. \end{matrix}

Also we define the “combinatorial translation from the walls”

T′ = T_{Γ}^{Ω} : K(Γ, Ŝ) → K(Ω, Ŝ)

by the prescriptions

(T′N)(λ) = N (λ_{Γ})

(T′N)(λ, β) = \{ \begin{matrix}(a_{λ}^{β} ⊕ 1)N (λ_{Γ}, β), & if β ↑ λ_{Γ} ≠ λ_{Γ}; \\ N (λ_{Γ}, β) ⊕ N (λ_{Γ} + pβ, β), & if (β ↑ λ)_{Γ} = λ_{Γ} + pβ; \\ \{(x + a_{λ}^{β}y, y) | x, y ∈ N (λ_{Γ}, β)\}, & if (β ↑ λ)_{Γ} = λ_{Γ} \\ and β ↑ λ ≠ λ. \end{matrix}

That these are the correct combinatorial translations is the content of the following Theorem 4.3.1. Suppose p is at least the Coxeter number. Then it is possible to simultaneously choose extensions

e^{β}(λ) ∈ Ext_{C}^{1}(Z^{β}(λ), Z^{β}(β ↑ λ))

for all λ ∈ X, β ∈ R^{+} with β ↑ λ ≠ λ such that for the functors 𝕍 defined by these choices we have natural equivalences

T_{Γ}^{Ω}𝕍_{Γ} ≅ 𝕍_{Ω}T_{Γ}^{Ω}, T_{Ω}^{Γ}𝕍_{Ω} ≅ 𝕍_{Γ}T_{Ω}^{Γ}

whenever Ω is regular and Γ is subregular (i.e. each element lies on only one wall) or Γ consists of special points.

The proof is a big calculation contained in [AJS94], 10–13. We will not repeat it here. Let now T^{out} denote the combinatorial translation functors from anything to the orbit W ·_{p} 0. We denote the combinatorial wall-crossings and their concatenations also by Θ_{s}, Θ_{σ}. From the theorem we deduce

BŜ = End_{K}(⨁ 𝕍Θ_{σ}T^{out}ZŜ(v)) = End_{K}(⨁ Θ_{σ}T^{out}ZŜ(v))

and this is indeed a purely combinatorial description of BŜ.

4.4. Independence of the characteristic. Let us now explain how one constructs the algebra whose existence is claimed in Theorem 4.1.2. To begin with let D be any integral domain with a morphism S(ℤ′) → D that doesn’t annihilate any h_{α}. We denote the image of h_{α} in D by h_{α} as well, form D^{∅} = D[h_{α}^{−1} | α ∈ R^{+}] and for any β ∈ R^{+} also D^{β} = D[h_{α}^{−1} | α ∈ R^{+}, α ≠ β]. Let in addition O be any orbit of facets under the affine Weyl group. Then we define the combinatorial category K(O, D) in analogy with our previous definition. So an object of K(O, D) consists of the following data:

1. A collection M(F), F ∈ O of finitely generated D^{∅}-modules, almost all of them zero.

2. For each β ∈ R^{+}, F ∈ O a finitely generated D^{β}-submodule M(F, β) of M(F) ⊕ M(β ↑ F) resp. M(F) if β ↑ F ≠ F resp. β ↑ F = F.

Here we put β ↑ F = s_{H}F with H = H_{β,m} for the minimal m such that F ⊄ H_{β,m}^{+}. It is clear that for Ω ⊂ X a (·_{p})-orbit contained in O we have K(O, Ŝ(k)) = K(Ω, Ŝ(k)) canonically. Let us for every special point v ∈ P denote by O(v) its orbit and define

Z_{D}(v) ∈ K(O(v), D)

by the formulas

(Z_{D}(v))(F) = \{ \begin{matrix}D^{∅} & if F = v; \\ 0 & else, \end{matrix}
(Z_{D}(v))(F, β) = \{ \begin{matrix}D^{β} & if F = v; \\ 0 & else. \end{matrix}

Remark in addition that a_{λ}^{β} is a well defined element of any D^{∅} and depends only on the alcove of λ and the orbit O of facets containing Γ, so given an orbit of facets O we can associate in the same way to any A ∈ A and β ∈ R^{+} the constant a_{A}^{β} ∈ D^{∅}. Thus our formulas for combinatorial translations give us functors

T_{A}^{O} : K(A, D) → K(O, D)

and

T_{O}^{A} : K(O, D) → K(A, D).

Again we define the wall-crossings

Θ_{s} = T_{s}^{A}T_{A}^{s} : K(A, D) → K(A, D),

their concatenations Θ_{σ} and the T^{out} : K(O, D) → K(A, D) for any O. Now we simply construct the algebra B_{S(ℤ′)} we were after as

B_{S(ℤ′)} = End_{K}(⨁ Θ_{σ}T^{out}Z_{S(ℤ′)}(v)).

To prove that it has the properties asserted in Theorem 4.1.2 for chark = 0 or chark ≫ 0 is a rather dull base change argument. Remark that this would already be enough for our application to Lusztig’s conjecture. To get the properties asserted for all k with chark ≥ h is much more subtle, but we have to omit the details contained in [AJS94], 14–16. Let us just remark that this involves putting ℤ-gradings on our K-categories and on B_{S(ℤ′)}. These seem to us a very interesting structure which should be explored further. Remark for example that from there we get ([AJS94], 18.17) quite remarkable ℤ-gradings on the restricted enveloping algebras U^{res}(𝔤) themselves that are not visible to the naked eye.

5. Corrections to [AJS94]

Here comes a list of the errors in [AJS94] we know up to now. We are grateful to Steen Ryom-Hansen and Masaharu Kaneda for pointing out some of them.

page 20, (D): … for all μ ∈ X, m ∈ … page 20, (C): … for all μν ∈ X. page 20, (B): … for all μ ∈ X. page 21, ↑ 4: ζ^{d_{α}⟨μ,α^{∨}⟩}

page 21, 8.13(2): 0 ⟶ Z^{β}(λ′) ⟶^{i} Y ⟶^{j} Z^{β}(λ) ⟶ 0

page 44, ↓ 19: The existence follows from 2.16.b page 61, ↑ 2: … from 5.7(3) to construct … page 62, ↓ 17: … we can take π̃(H_{α}) … page 69, ↓ 9: … chZ_{F} (λ − n_{β}(λ)β) … page 69, ↓ 11: … L_{F} (λ − n_{β}(λ)β) … page 70, ↓ 12: … the definition in 6.9 and 6.10. page 73, ↓ 5: … (cf. 1.3) … page 78, ↑ 12: … E_{w·μ−λ} = E_{w′(μ−λ)}, where w′ is the linear part

of w. page 100, ↑ 4,6: Z_{A} ersetzen durch Z_{β} page 104, ↓ 15: for all λ ∈ X. page 108, ↑ 5: Hom_{C}(Z^{β}(β ↑ λ), M^{β} = 0, … page 113, ↑ 12, 14: (2) wird (3) und (3) wird zu (4) page 124, ↓ 7: by (3); we have obviously page 130, Theorem ↑ 5: … there is exactly one H ∈ H(β) with

α(H) = β and μ ∈ H; page 130, 15.3: In the part preceding the formula (1) all three <

should be > and all three > should be < . page 131, ↓ 21: … as in 7.5. page 189, ↓ 11: … Set for all μ ∈ Ω

page 191, ↓ 6, (4): \{ \begin{matrix}Hom_{𝔤}(M_{k}, N_{k}), … \\ Hom_{u}(M_{k}, N_{k}), … \end{matrix}

page 192, ↓ 5, (5): … if (β ↑ λ)_{Γ} = Γ. page 192, ↑ 14: change 10.12 into 10.2(1). page 201, ↑ 8: … FC_{A(k)}(Ω), … page 205, ↓ 6: … (β ↑ λ)_{Γ} ∈ β … page 211, ↓ 13: … S_{k}^{β}(0, h_{β}g_{i}) … page 213, 16: In the quantum group case one should add another

assumption to most results in Section 16 involving a ♯. The problem is that in the proof of part two of Proposition 16.5 c) we use 14.13(4) and thus need that p(k) is prime to the connection index [X : ℤR] or, more generally, the following condition (∗): Ω_{k} ∩ (Ω_{k} + pν) ≠ ∅ for ν ∈ X implies ν ∈ ℤR. Under the hypothesis p(k) > h this condition is always satisfied for the choice λ_{0}(k) = 0, i.e. for the “principal block”. Indeed, for p(k) > h the equality wρ−ρ = pν with w ∈ W and ν ∈ X implies already ν = 0. The condition (∗) should be added to make true the statements involving an upper index ♯ in Proposition 16.5 c), 16.8(4), 16.9(3), and Theorem 16.18.

page 215, ↓ 16: (Recall from 9.7 that …) page 222, last line: Both λ should be ν. page 224,16.19(1): Here we should remark that in our application

we know N_{1} by 16.11, so the place where we loose control for small characteristic is rather the choice of N_{2} and N around 16.19(2) and (3).

page 235, ↓ 17: … Lemma 15.17 implies that… page 237, ↓ 1: a direct summand of Q^{(λ)}(S_{k}) … page 254: In the last big displayed calculation of 18.15 four lower

indices Γ are missing. The correct formula is

S_{k}⟨−o(λ, Γ)⟩ ≅ Hom_{E_{Ω},Y} (T^{∗}Z̃_{S_{k}}(λ_{Γ}), Z̃′_{S_{k}}(λ))
≅ Hom_{E_{Γ},Y} (Z̃_{S_{k}}(λ_{Γ}), T_{∗}Z̃′_{S_{k}}(λ))
≅ Hom_{E_{Γ},Y} (Z̃_{S_{k}}(λ_{Γ}), T_{!}Z̃′_{S_{k}}(λ)⟨i⟩)
≅ Hom_{E_{Γ},Y} (Z̃_{S_{k}}(λ_{Γ}), Z̃′_{S_{k}}(λ_{Γ})⟨−o(λ, Γ) + i⟩)
≅ S_{k}⟨i − o(λ, Γ)⟩.

page 283-296, Appendices B and C: In Appendix B we work with the version of the braid group action introduced in [Lus93]. In Appendix C we work with the old version from [Lus90b]. The two versions differ on each weight space by a sign, so the root vectors constructed with the different versions differ by a sign. So if Proposition C.5 and Corollary C.6 hold for one version, they also hold for the other.

page 309: The last M in line 11 should be barred. page 317, ↑ 13: … (Contemporary Mathematics …)

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Universität Freiburg, Mathematisches Institut, Albertstraße 23b, D-79104 Freiburg, Germany

  E-mail address: soergel@sun1.mathematik.uni-freiburg.de