CHARACTER FORMULAS FOR TILTING MODULES OVER KAC-MOODY ALGEBRAS

WOLFGANG SOERGEL

Abstract. We show how to express the characters of tilting modules in a (possibly parabolic) category O over a Kac-Moody algebra in terms of the characters of simple highest weight modules. This settles in lots of cases Conjecture 7.2 of [Soe97] describing the character of tilting modules for quantum groups at roots of unity.

Introduction

In this article I determine the characters of indecomposable tilting modules in the category O over an affine Kac-Moody Lie algebra. Via an equivalence of categories due to Kazhdan and Lusztig this leads to character formulas for tilting modules over quantum groups, in particular we prove conjecture 7.2 from [Soe97] in many cases.

I found the key to the determination of these characters in the preprint [Ark96]. There Arkhipov extends Feigin’s semi-infinite cohomology and shows in particular, that the category of all modules with a Weyl filtration in positive level is contravariantly equivalent to the analogous category in negative level. Under this equivalence projective objects have to be transformed into tilting modules, thus the Kazhdan-Lusztig conjectures in positive level leads to character formulas for tilting modules in negative level.

In [Ark96] the contravariant equivalence alluded to above appears as illustration of a much stronger and deeper semi-infinite duality. I will show in the subsequent sections, how one can get it directly. Then I will discuss the application to tilting modules.

1. The semi-regular bimodule

Let 𝔤 = ⨁_{i∈ℤ} 𝔤_{i} be a ℤ-graded Lie algebra over the field k with finite dimensional homogeneous pieces, dim_{k} 𝔤_{i} < ∞ for all i.

Definition 1.1. A character γ : 𝔤_{0} → k is called a semi-infinite character for 𝔤 iff we have:

1. As a Lie algebra 𝔤 is generated by 𝔤_{1}, 𝔤_{0} and 𝔤_{−1}.

1991 Mathematics Subject Classification. 17B70, 17B67, 17B37.

2. γ([X, Y]) = tr(adXadY : 𝔤_{0} → 𝔤_{0}) ∀X ∈ 𝔤_{1}, Y ∈ 𝔤_{−1}.

Remark 1.2. Voronov [Vor93] and Arkhipov [Ark96] work without the assumption (1) and rather ask of γ, that dγ is the “critical cocycle of 𝔤”. The Lie algebras we are interested in however are generated by 𝔤_{1}, 𝔤_{0} and 𝔤_{−1}. Since it is easier to treat this case, we restrict our attention to the framework mapped out by the above definition.

We put 𝔟 = 𝔤_{≥0}, 𝔫 = 𝔤_{<0} and denote the enveloping algebras of 𝔤, 𝔟, 𝔫 by U, B, N. Certainly U, B and N inherit a ℤ-grading from the corresponding Lie algebras. We consider for N the graded dual N^{⊛} = ⊕_{i}N_{i}^{∗} and make it into an N-bimodule via the prescriptions (fn)(n_{1}) = f(nn_{1}), (nf)(n_{1}) = f(n_{1}n) ∀ f ∈ N^{⊛}, n, n_{1} ∈ N.

Theorem 1.3. Let γ : 𝔤_{0} → k be a semi-infinite character for 𝔤. Then there exists a ℤ-graded U-bimodule S = S_{γ} along with an inclusion of ℤ-graded N-bimodules ι : N^{⊛} ↪ S such that the following hold:

1. The map U ⊗_{N} N^{⊛} → S, u ⊗ f ↦ uι(f) is a bijection.

2. The map N^{⊛} ⊗_{N} U → S, f ⊗ u ↦ ι(f)u is a bijection.

3. Up to a twist by γ the inclusion ι : N^{⊛} ↪ S commutes with the adjoint action of 𝔤_{0} on both spaces, in formulas ι(f ∘ adH) + (adH)ι(f) = ι(f)γ(H) for all H ∈ 𝔤_{0} und f ∈ N^{⊛}.

Remarks 1.4. 1. The bimodule S is a semi-infinite analogue of the enveloping algebra, since in analogy to U ≅ B ⊗_{k} N ≅ N ⊗_{k} B we have S ≅ B ⊗_{k} N^{⊛} ≅ N^{⊛} ⊗_{k} B. We call S the “semiregular bimodule”. In greater generality it is introduced in [Vor93] as “standard semijective module” and in [Ark96] as “semiregular module”. The presentation in [Vor93] however still needs some fixing.

2. The semi-regular bimodule S only has homogeneous components of degree ≥ 0.

3. The formula (3) in particular tells us that Hι(ε) = ι(ε)(H +γ(H)) for all H ∈ 𝔤_{0} and ε ∈ N_{0}^{⊛}, in particular for ε the augmentation of N.

4. Going carefully through the proof of the theorem and replacing the comparision of dimension at the end by a more refined argument we see that it is sufficient to assume dim_{k} 𝔤_{i} < ∞ for i < 0.

Proof. Let us start by constructing for an arbitrary character γ of 𝔤_{0} a vector space S = S_{γ} with a left and a right action of U.

For any two ℤ-graded Vector spaces M, M′ let us define the ℤ-graded Vector space Hom_{k}(M, M′) with homogeneous components

Hom_{k}(M, M′)_{j} = {f ∈ Hom_{k}(M, M′) | f(M_{i}) ⊂ M′_{i+j}}.

For example we have N^{⊛} = Hom_{k}(N, k), if we equip k with the ℤ-Grading k = k_{0}. Now let us consider for an arbitrary character γ of 𝔤_{0} the following sequence of isomorphisms of ℤ-graded vector spaces over k:

Hom_{B}(U, k_{γ} ⊗_{k} B) →^{∼} Hom_{k}(N, B) ←^{∼} N^{⊛} ⊗_{k} B →^{∼} N^{⊛} ⊗_{N} U

Here Hom_{B} denotes the space of all B-homomorphismen in Hom_{k}, our k_{γ} is the one-dimensional representation of 𝔟 given by the character γ : 𝔤_{0} → k and the surjection 𝔟 ↠ 𝔤_{0}, and k_{γ} ⊗_{k} B is as a left 𝔟-module the tensor product of these two representations. The first isomorphism is defined as the restriction to N using the identification k_{γ} ⊗_{k} B →^{∼} B, 1 ⊗ b ↦ b. The other isomorphisms are obvious. We now put S_{γ} = N^{⊛} ⊗_{k} B and define on this space an action of U from the left (resp. right) via the first two (resp. the last) isomorphisms. Our first goal is to show, that for a semi-infinite character γ the right and the left action of U on S_{γ} commute. I have to confess that I don’t understand the true reason for that and thus have to check by blind calculation.

All our isomorphisms above are compatible with the obvious left action of N and right action of B on our spaces. Thus the left action of N commutes with the right action of U, and similarly the right action of B commutes with the left action U. We thus only have to show that

\begin{aligned}H((f ⊗ b)Y) & = & (H(f ⊗ b))Y, \\ X((f ⊗ b)Y) & = & (X(f ⊗ b))Y \end{aligned}

for all H ∈ 𝔤_{0}, X ∈ 𝔤_{1}, Y ∈ 𝔤_{−1}, f ∈ N^{⊛}, b ∈ B. Here we may even assume b = 1. Indeed, by right multiplication with X_{1} ∈ 𝔤_{1} resp. H_{1} ∈ 𝔤_{0} and a short calculation one easily deduces from our equations with b the analogous equations with bX_{1} resp. bH_{1}. Thus we need only to show

\begin{aligned}H((f ⊗ 1)Y) & = & (H(f ⊗ 1))Y, \\ X((f ⊗ 1)Y) & = & (X(f ⊗ 1))Y \end{aligned}

for all H ∈ 𝔤_{0}, X ∈ 𝔤_{1}, Y ∈ 𝔤_{−1}, f ∈ N^{⊛}.

So let’s calculate. If we denote by L_{Y} : N → N the multiplication with Y ∈ 𝔤_{−1} from the left, we have (f ⊗ 1)Y = fL_{Y} ⊗ 1. Also for H ∈ 𝔤_{0} we get from our definitions

H(f ⊗ 1) = −f(adH) ⊗ 1 + f ⊗ (γ(H) + H)

where adH : N → N is given by (adH)(n) = Hn − nH as usual. Thus we get

(H(f ⊗ 1))Y = −f(adH)L_{Y} ⊗ 1 + fL_{Y} ⊗ (γ(H) + H),
+fL_{[H,Y]} ⊗ 1,
H((f ⊗ 1)Y) = −fL_{Y} (adH) ⊗ 1 + fL_{Y} ⊗ (γ(H) + H),

and since (adH)L_{Y} = L_{Y} (adH)+L_{[H,Y]} these expressions still coincide for γ arbitrary.

To determine X(f ⊗ 1) we choose a basis (H_{i})_{i∈I} of 𝔤_{0} and define linear maps H_{X}^{i}, F_{X} : N → N by

nX = Xn + ∑_{i} H_{i}H_{X}^{i}(n) + F_{X}(n) ∀n ∈ N.

From our definitions we get

X(f ⊗ 1) = f ⊗ X + ∑_{i} fH_{X}^{i} ⊗ (γ(H_{i}) + H_{i}) + fF_{X} ⊗ 1.

We further calculate

(Y n)X = X(Y n) + ∑ H_{i}H_{X}^{i}(Y n) + F_{X}(Y n)
Y (nX) = Y Xn + ∑ Y H_{i}H_{X}^{i}(n) + Y F_{X}(n)
= XY n + [Y, X]n + ∑ H_{i}Y H_{X}^{i}(n)
+ ∑[Y, H_{i}]H_{X}^{i}(n) + Y F_{X}(n)

and with [Y, X] = ∑ c_{Y X}^{i}H_{i} we get the formulas

\begin{aligned}H_{X}^{i}L_{Y} & = & L_{Y} H_{X}^{i} + c_{Y X}^{i}id_{N} \\ F_{X}L_{Y} & = & L_{Y} F_{X} + ∑_{i} L_{[Y,H_{i}]}H_{X}^{i}. \end{aligned}

Now we get

X((f ⊗ 1)Y) = fL_{Y} ⊗ X + ∑_{i} fL_{Y} H_{X}^{i} ⊗ (γ(H_{i}) + H_{i})
+fL_{Y} F_{X} ⊗ 1
(X(f ⊗ 1))Y = f ⊗ XY + ∑_{i} fH_{X}^{i} ⊗ (γ(H_{i}) + H_{i})Y + fF_{X} ⊗ Y
= fL_{Y} ⊗ X + f ⊗ [X, Y]
+ ∑_{i} fH_{X}^{i}L_{Y} ⊗ (γ(H_{i}) + H_{i})
+ ∑_{i} fH_{X}^{i}L_{[H_{i},Y]} ⊗ 1
+fF_{X}L_{Y} ⊗ 1

as a condition for X((f⊗1)Y) = (X(f⊗1))Y we find using our formulas 0 = ∑_{i} c_{Y X}^{i} · γ(H_{i}) + c_{[H_{i},Y]X}^{i}, in other words γ([X, Y]) = tr(adXadY : 𝔤_{0} → 𝔤_{0}). Thus for every semi-infinite character γ our S = S_{γ} is a U-bimodule. We define ι : N^{⊛} ↪ S by ι(f) = f ⊗ 1 and only have to check the properties (1)–(3) from the theorem.

Here (2) and (3) follow directly from the definitions and we only prove (1). Certainly S admits a ℤ-grading with finite dimensional homogeneous components. By equality of dimensions it is sufficient to show that U ⊗_{N} N^{⊛} → S is a surjection, hence that ι(N^{⊛}) already generates S as a left U-module. But our formulas for X(f ⊗1), H(f ⊗1) show, that the left U-submodule generated by ι(N^{⊛}) is stable under the right B-action.

2. The category of all modules with a finite Verma-flag is its own opposed category

We keep the notations of the preceding section. Let γ : 𝔤_{0} → k be a semi-infinite character for 𝔤 and S = S_{γ} the corresponding semiregular bimodule. Let M resp. K denote the categories of all ℤ-graded representations of 𝔤, which are over N graded free resp. cofree of finite rank, i.e. isomorphic to finite direct sums of may-be grading shifted copies of N resp. N^{⊛}.

Theorem 2.1. The functor S⊗_{U} defines an equivalence of categories S⊗_{U} : M →^{∼} K, such that short exact sequences correspond to short exact sequences.

Remarks 2.2. 1. The existence of an equivalence M ≅ K is a result

of Arkhipov [Ark96].

2. If 𝔥 ⊂ 𝔤_{0} is an abelian subalgebra such that the adjoint action of 𝔥 on 𝔫 is diagonalizable, then our functor also gives an equivalence M_{𝔥} →^{∼} K_{𝔥} between the categories of all 𝔥-diagonalizable objects of M resp. K.

Proof. First we deduce from S ≅ N^{⊛}⊗_{N} U, that S⊗_{U} ≅ N^{⊛}⊗_{N} indeed gives a functor T : M → K, which transforms short exact sequences into short exact sequences. Furthermore multiplication from the right defines an isomorphism N^{opp} → End_{N}(N^{⊛}), and since we also have S ≅ U ⊗_{N} N^{⊛} the prescription Hom_{U}(S, ) ≅ Hom_{N}(N^{⊛}, ) indeed defines a functor H : K → M making short exact sequences to short exact sequences.

Our functors obviously form an adjoint pair (T, H). To show they are inverse equivalences of categories, we only have to show that for all M ∈ M resp. K ∈ K the canonical map M → HT M resp. T HK → K is an isomorphism. But we have

\begin{aligned}HT M & = & Hom_{U}(S, S ⊗_{U} M) \\ & = & Hom_{N}(N^{⊛}, N^{⊛} ⊗_{N} M) \end{aligned}

and for a free N-module M of finite rank certainly the canonical map M → Hom_{N}(N^{⊛}, N^{⊛} ⊗_{N} M) is an isomorphism. Similarly we have

\begin{aligned}T HK & = & S ⊗_{U} Hom_{U}(S, K) \\ & = & N^{⊛} ⊗_{N} Hom_{N}(N^{⊛}, K) \end{aligned}

and for K = N^{⊛} or, more generally, K cofree of finite rank over N certainly the canonical map N^{⊛} ⊗_{N} Hom_{N}(N^{⊛}, K) → K is an isomorphism.

In the following corollary the content of the theorem appears most clearly. For a ℤ-graded space V = ⊕V_{i} let V^{⊛} = Hom_{k}(V, k) denotes its ℤ-graded dual with homogeneous components (V^{⊛})_{i} = (V_{−i})^{∗}. If V is a ℤ-graded representation of 𝔤, the contravariant action (Xf)(v) = −f(Xv) for all f ∈ V^{⊛}, X ∈ 𝔤, v ∈ V makes V^{⊛} into a ℤ-graded representation of 𝔤. Let M^{opp} denote the opposed category of M.

Corollary 2.3. The functor M ↦ (S⊗_{U}M)^{⊛} defines an equivalence of categories M →^{∼} M^{opp}, under which short exact sequences correspond to short exact sequences, and such that U ⊗_{B} E gets mapped to U ⊗_{B} (k_{−γ} ⊗ E^{∗}), for every finite dimensional ℤ-graded representation E of 𝔤_{0}.

Proof. Remark that the formulas above define a second left action of 𝔫 on N^{⊛} that doesn’t coincide with the left action from section 1 in general. However N^{⊛} with this second 𝔫-action is isomorphic to N^{⊛} with the first action as a ℤ-graded 𝔫-module, a possible isomorphism being the transpose of the principal antiautomorphism of N. Hence our functor V ↦ V^{⊛} defines an equivalence of categories K →^{∼} M^{opp}. The rest of the proof is left to the reader.

Remarks 2.4. 1. It is not difficult to show that M consists precisely of those ℤ-graded 𝔤-modules, which admit a finite filtration with subquotients of the form U ⊗_{B} E for suitable finite dimensional irreducible ℤ-graded 𝔤_{0}-modules E . Therefore we call M also the category of all finite Verma flag modules.

2. It is also not difficult to show that M is stable under taking direct summands. More generally an arbitrary direct summand of a graded free N-module of finite rank is itself graded free of finite rank, for any ℤ-graded algebra N which has no terms of positive degree and whose degree zero part is just the ground field.

3. Under the assumptions of remark 2.2 (2) our functor also gives an equivalence M_{𝔥} →^{∼} M_{𝔥}^{opp}.

3. Projective objects in O

We now develop some well-known results in great generality and need stronger assumptions than in the first sections. Let from now on k be an algebraically closed field of characteristic zero and let 𝔤 = ⨁_{i∈ℤ} 𝔤_{i} be a ℤ-graded Lie algebra over k with dim_{k} 𝔤_{i} < ∞ ∀i ∈ ℤ such that 𝔤_{0} is reductive and 𝔤 a semisimple 𝔤_{0}-module for the adjoint action. Then we consider the category O of all ℤ-graded 𝔤-modules M = ⨁_{i∈ℤ} M_{i}, which are locally finite for 𝔤_{≥0} and semisimple for 𝔤_{0}. For M, N ∈ O we thus have Hom_{O}(M, N) = {f ∈ Hom_{𝔤}(M, N) | f(M_{i}) ⊂ N_{i} ∀i ∈ ℤ}.

As before we put 𝔤_{≥0} = 𝔟 and U(𝔟) = B. Let Λ denote the set of isomorphism classes of irreducible finite dimensional ℤ-graded 𝔤_{0}- modules. Such an E ∈ Λ certainly will be concentrated in one degree |E| ∈ ℤ, so we have E = E_{|E|}. For E ∈ Λ we form the Verma module Δ(E) = U ⊗_{B} E. Certainly Δ(E) is an object of O, has a unique simple quotient L(E), and {L(E)}_{E∈Λ} is a system of representatives for the simple O. Dually we form the object ∇(E) = Hom_{𝔤_{≤0}}(U, E) in O and it is easy to see that L(E) is the smallest non-zero submodule of ∇(E). More precisely we consider for n ∈ ℤ in O the subcategory

O_{≤n} = {M ∈ O | M_{i} = 0 if i > n}.

One shows for arbitrary E, F ∈ Λ:

Lemma 3.1. 1. Δ(E) is the projective cover of L(E) in O_{≤|E|}. 2. ∇(E) is the injective hull of L(E) in O_{≤|E|}.

3. Hom_{O}(Δ(F), ∇(E)) = 0 if F ≠ E. For F = E this space is onedimensional.

4. Ext_{O}^{1}(Δ(F), ∇(E)) = 0.

Proof. Left to the reader.

In general a simple object does not admit a projective cover in O, but only in the truncated categories O_{≤n}.

Theorem 3.2. 1. Every simple object L(E) ∈ O_{≤n} admits in O_{≤n} a projective cover P_{≤n}(E), and this projective cover has a finite Δ-flag.

2. For m > n the kernel of the surjection P_{≤m}(E) ↠ P_{≤n}(E) admits a finite Δ-flag with subquotients of the form Δ(F) for m ≥ |F | > n.

3. The simple L(E) admits a projective cover P (E) in O if and only if P_{≤n}(E) ≅ P_{≤n+1}(E) ≅ … for n ≫ 0, and then we have P (E) ≅ P_{≤n}(E).

The proof needs some abstract theory.

Lemma 3.3. Let A be an abelian category, p : P ↠ L a surjection of an indecomposable projective object onto a simple object. If E = End_{A}P is of finite length as a right module over itself, then P is a projective cover of L.

Proof. Indeed zero and one are the only idempotents of E, since P is indecomposable. The usual arguments via the Fitting decomposition then show, that every element of E is either nilpotent or invertible. Now if i : U → P is a morphism such that p∘i ≠ 0, then by projectivity of P there exists q : P → U such that p ∘ i ∘ q = p. But then i ∘ q is not nilpotent, hence an isomorphism and thus i is surjective. Hence ker p is the biggest proper subobject of P, what was to be shown.

Now we prove the Theorem.

Proof. (1) For a ℤ-graded 𝔟-module K = ⨁ K_{i} let τ_{≤n}K denote the quotient by the submodule of all homogeneous parts of degree > n, thus τ_{≤n}K = ⨁_{i≤n} K_{i}. For E ∈ Λ then

Q = U ⊗_{B} τ_{≤n}(B ⊗_{U(𝔤_{0})} E)

is projective in O_{≤n}. Indeed, if Hom_{𝔤,ℤ} denotes the space of all 𝔤module homomorphisms which are homogeneous of degree zero with respect to the ℤ-grading, then

Hom_{O}(Q, M) = Hom_{𝔤,ℤ}(Q, M)
= Hom_{𝔟,ℤ}(τ_{≤n}(B ⊗_{U(𝔤_{0})} E), M)
= Hom_{𝔟,ℤ}(B ⊗_{U(𝔤_{0})} E, M)
= Hom_{𝔤_{0},ℤ}(E, M)

for all M ∈ O_{≤n} and Hom_{𝔤_{0},ℤ}(E, M) is an exact functor in M ∈ O by the very definiton of this category.

Certainly Q graded free over N of finite rank, and if n ≥ |E| there is a surjection Q ↠ L(E). By 3.3 we can take as P_{≤n}(E) every indecomposable summand of Q which has L(E) as a quotiente, and by remark 2.4 (2) the module P_{≤n}(E) admits a Δ-flag.

(2) Certainly for m ≥ n we have a surjection

P_{≤m}(E) ↠ P_{≤n}(E),

and by the universal properties the kernel of such a surjection has to be submodule of P_{≤m}(E) generated by all homogeneous components of degree > n. But such a submodule in a graded free N-module is graded free itself (with a basis all vectors of degree > n inside a homogeneous basis of the full module), thus by remark 2.4 (2) the submodule admits a Δ-flag, too.

(3) If L(E) admits a projective cover P (E) in O, then P (E) is generated by a single vector (indeed by every vector outside the biggest proper submodule), thus P (E) ∈ O_{≤n} for n ≫ 0, hence P (E) ≅ P_{≤m}(E) ∀m ≥ n.

If on the other hand the projective system of the P_{≤n}(E) stabilizes with an object P (E), we have to show that P (E) is projective in O. Certainly P (E) is generated by one element v. Now let M ↠ M′ be a surjection and f′ : P (E) → M′ a morphism, which we want to lift to f : P (E) → M. Then we choose a preimage m ∈ M of m′ = f′(v) ∈ M′ and consider the surjection U(𝔤)m ↠ U(𝔤)m′. Now both module lie in O_{≤n} for n ≫ 0, and since P (E) ≅ P_{≤n}(E) we find the lift we were looking for.

4. Reciprocity and decompositon of O into blocks

To formulate the usual reciprocity in full generality we have to introduce multiplicities in full generality.

Definition 4.1. Let A be an abelian category, M ∈ A an object, L ∈ A a simple object. The multiplicity [M : L] ∈ ℕ ∪ {∞} of L in M is the supremum over all (finite) filtrations F of M of the multiplicity of L as a subquotient of the filtration. In formulas

[M : L] = sup_{F} #{i | F_{i}M/F_{i+1}M ≅ L}.

This multiplicity is additive, i.e. for every short exact sequence M′ ↪ M ↠ M′′ we have [M : L] = [M′ : L] + [M′′ : L]. In particular for ∇ ∈ O_{≤n} and E ∈ Λ we always have [∇ : L(E)] = dim_{k} Hom_{O}(P_{≤n}(E), ∇). On the other hand for a module P ∈ O with a finite Δ-flag we know by Lemma 3.1 that the multipicity [P : Δ(F)] of Δ(F) as a subquotient is dim_{k} Hom(P, ∇(F)). Putting things together we obtain the reciprocity formula

[P_{≤n}(E) : Δ(F)] = [∇(F) : L(E)]

for all E, F ∈ Λ und n ≥ max{|E|, |F |}.

Since it is not a big deal from where we are, I want to discuss the decomposition of O into blocks, although it is not needed for the main result of this article. Certainly there is a partial order ≥ on Λ such that [Δ(F) : L(E)] ≠ 0 ⇒ F ≥ E und [∇(F) : L(E)] ≠ 0 ⇒ F ≥ E. Let from now on ≥ be the smallest such partial order and ∼ the equivalence relation generated by it. For every equivalence class θ ∈ Λ/∼ we consider the category

O_{θ} = {M ∈ O | [M : L(E)] ≠ 0 ⇒ E ∈ θ}.

Theorem 4.2. The functor ∏_{θ∈Λ/∼} O_{θ} → O, (M_{θ})_{θ} ↦ ⨁_{θ} M_{θ} is an equivalence of categories.

Proof. For θ ∈ Λ/∼ and M ∈ O let M_{θ} ⊂ M denote the submodule generated by all images of morphisms φ : P_{≤n}(E) → M with E ∈ Λ. We show

1. M_{θ} ∈ O_{θ};

2. If f : M → N is a morphism in O, then f(M_{θ}) ⊂ N_{θ};

3. M = ⨁_{θ} M_{θ}.

This then proves the theorem. We start with (1). By the definition of our equivalence relation and by the reciprocity formula we have P_{≤n}(E) ∈ O_{θ} for all E ∈ θ. In a short exact sequence in O the middle term lies in O_{θ} iff both ends do, by the additivity of multiplicities. Thus it will be sufficient to show that O_{θ} is stable under arbitrary direct sums. But every simple subquotient of ⨁_{i∈I} M_{i} is the quotient of a submodule generated by one element, thus occurs also in a finite direct sum. This establishes (1). Now (2) follows from the definitons and from (2) we see that the sum in (3) is direct. We leave it to the reader to show it is all of M.

Remark 4.3. Let E ∈ Λ be given. If there are only finitely many F ∈ Λ such that F ≥ E, then L(E) admits by theorem 3.2 (3) a projective cover P (E) in O.

5. Tilting modules in O

Definition 5.1. By a Δ-flag in an object M ∈ O we mean a (possibly infinite) increasing filtration

0 = M_{0} ⊂ M_{1} ⊂ M_{2} ⊂ …

such that M = ⋃ M_{ν} and M_{ν}/M_{ν−1} ≅ Δ(F_{ν}) for all ν ≥ 1, with suitable F_{ν} ∈ Λ.

Theorem 5.2. For every E ∈ Λ there is a unique up to isomorphism indecomposable object T = T (E) ∈ O such that:

1. Ext_{O}^{1}(Δ(F), T) = 0 for all F ∈ Λ;

2. T admits a Δ-flag, starting with T_{1} ≅ Δ(E).

Definition 5.3. This object T (E) is called the tilting module with parameter E.

Remark 5.4. The theorem is a variation of results of [Rin91], who in turn develops results of [Don86] and [CI89] in a general context.

Proof. We start with

Lemma 5.5. 1. For all F, G ∈ Λ both spaces Hom_{O}(Δ(F), Δ(G))

and Ext_{O}^{1}(Δ(F), Δ(G)) are of finite dimension.

2. For all G ∈ Λ and i ∈ ℤ there are at most finitely many F ∈ Λ such that |F | = i and Hom_{O}(Δ(F), Δ(G)) or Ext_{O}^{1}(Δ(F), Δ(G)) is not zero.

Proof. By our assumptions all homogeneous components of Δ(G) are of finite dimension. This gives (1) and (2) for Hom. Let furthermore n be bigger than |F | and |G|. We consider the short exact sequence ker ↪ P_{≤n}(F) ↠ Δ(F) and get a surjection

Hom_{O}(ker, Δ(G)) ↠ Ext_{O}^{1}(Δ(F), Δ(G)).

Since here ker admits a finite Δ-flag, we get (1) for Ext^{1}. Finally all Δsubquotients of a Δ-flag of P_{≤n}(F) are of the form Δ(H), where H is a summand of the ℤ-graded 𝔤_{0}-module (τ_{≤n−|F |}U(𝔤_{>0}))⊗_{k} F (and 𝔤_{0} acts by the adjoint action on the first factor). For Hom_{O}(ker, Δ(G)) to be different from zero, such an H also has to occur as a summand in the ℤgraded 𝔤_{0}-module Δ(G) = U(𝔤_{<0})⊗_{k}G, hence in (τ_{≥|F |−|G|}U(𝔤_{<0}))⊗_{k}G.

But the representation theory of reductive Lie algebras tells us, that for given finite dimensional representations U_{1}, U_{2} and G there are up to isomorphism only finitely many simple finite dimensional representations F such that U_{1} ⊗_{k} F and U_{2} ⊗_{k} G admit a common composition factor.

We show next

Proposition 5.6. Let E ∈ Λ, m ≤ |E|. There is a unique up to isomorphism indecomposable object T = T_{≥m}(E) in O such that

1. Ext_{O}^{1}(Δ(F), T) = 0 for all F ∈ Λ with |F | ≥ m;

2. There is an inclusion Δ(E) ↪ T, whose cokernel admits a finite Δ-flag such that only subquotients Δ(F) with |E| > |F | ≥ m occur.

Proof. We start by proving unicity. Let T′ be a second object satisfying our conditions. We consider the diagram

Δ(E) ↪ T ↠ coker
∥
Δ(E) ↪ T′ ↠ coker′.

Since by (2) for T′ and (1) for T the relevant Ext-group vanishes, we find α : T′ → T making the whole diagram commutative. Similarily we find β : T → T′. But α ∘ β isn’t nilpotent, hence an isomorphism since T was indecomposable with dim_{k}(End_{O}T) < ∞. The same holds for β ∘ α, and we deduce T ≅ T′.

Next we show the existence of T_{≥m}(E) by induction on m from above. As a basis for our induction we may take T_{≥|E|}(E) = Δ(E). Now let T_{≥m}(E) be constructed already. We then form a sequence T^{(i)} of objects from O as follows. Start with T^{(0)} = T_{≥m}(E). If T^{(i)} is already constructed and there is a nonsplit extension T^{(i)} ↪ T^{(i+1)} ↠ Δ(F) with F ∈ Λ, |F | = m − 1, take it as T^{(i+1)}. Otherwise stop at T^{(i)}.

We show that such a sequence T^{(i)} stops with a possible T_{≥m−1}(E). Indeed by Lemma 5.5 above the number

e(T) = ∑_{F ∈Λ, |F |=m−1} dim_{k} Ext_{O}^{1}(Δ(F), T)

is finite for all T ∈ O with finite Δ-flag. Now if |F | = |G| the Ext-group Ext_{O}^{1}(Δ(F), Δ(G)) disappears, and this gives us e(T^{(i+1)}) = e(T^{(i)}) − 1. Thus our sequence stops with an object T^{(j)} satisfying certainly conditions (1) and (2) for T_{≥m−1}(E) from the Proposition. Instead of proving that T^{(j)} is indeed indecomposable, it is easier to choose an indecomposable summand of T^{(j)}, whose homogeneous component of degree |E| doesn’t vanish, and this then is the possible T_{≥m−1}(E) looked for.

Proposition 5.7. Let E ∈ Λ and |E| ≥ n > m. Then there exists an inclusion T_{≥n}(E) ↪ T_{≥m}(E), and the cokernel of every such inclusion admits a Δ-flag with only subquotients Δ(F) for n > |F | ≥ m.

Proof. Consider in T_{≥m}(E) the submodule T′ generated by all homogeneous elements of degree at least n and form a short exact sequence T′ ↪ T_{≥m}(E) ↠ koker. Then T′ resp. koker admits a Δ-flag, in which there are only subquotients Δ(F) with |F | ≥ n resp. n > |F | ≥ m. It follows easily that T′ satisfies all conditions we put on T_{≥n}(E) except perhaps indecomposability. Thus T_{≥n}(E) is a direct summand of T′, and we deduce [T_{≥n}(E) : Δ(F)] ≤ [T_{≥m}(E) : Δ(F)]. But our inductive construction of T_{≥m}(E) shows the reverse inequality as well, for |F | ≥ n. Thus we have equality and can deduce T′ ≅ T_{≥n}(E). The Proposition follows.

After these preparations we can at least construct a possible T = T (E) as T = lim_{n→−∞} T_{≥n}(E). Let us remark right away that every element of EndT is either nilpotent or an isomorphism, since T is indecomposable and all its homogeneous components are finite dimension, so the usual arguments via the Fitting-decomposition work. To prove unicity we use

Lemma 5.8. In O there are enough injectives.

Proof. For M ∈ O the biggest ℤ-graded 𝔤-submodule of Hom_{𝔤_{0}}(U, M) which lies in O is an injective object of O containing M.

Remark 5.9. By general arguments [HS71] I.9.2 we can deduce that every simple object L(E) in O admits an injective hull I(E). One may show in addition that I(E) admits a (possibly infinite) increasing ∇- flag, whose multiplicities are given by the reciprocity formula [I(E) : ∇(F)] = [Δ(F) : L(E)]. We will neither use nor prove this.

Lemma 5.10. Let J ∈ O be such that Ext_{O}^{1}(Δ(F), J) = 0 ∀F ∈ Λ. Then also Ext_{O}^{1}(M, J) = 0 for all M ∈ O with a Δ-flag.

Proof. Choose a short exact sequence J ↪ I ↠ K with I injective in O. We get an exact sequence

Hom(M, J) ↪ Hom(M, I) → Hom(M, K) ↠ Ext^{1}(M, J),

which starts with an inclusion and ends with a surjection. Let 0 = M_{0} ⊂ M_{1} ⊂ … be a Δ-flag of M. Then we may rewrite the first three terms of our sequence to

lim_{←} Hom(M_{i}, J) ↪ lim_{←} Hom(M_{i}, I) → lim_{←} Hom(M_{i}, K),

and by our assumption we have for every fixed i a short exact sequence. Using our assumption on J a second time, we see that in addition all maps of the projective system Hom(M_{i}, J) are surjections. Therefore by [AM69], 10.2 the projective limit of our short exact sequence is a short exact sequence too and thus Ext^{1}(M, J) = 0.

We now show unicity of T = T (E). Let T′ ∈ O be a second object satisfying all the conditions of the theorem. As before we consider the diagram

Δ(E) ↪ T ↠ T/T′_{1}
∥
Δ(E) ↪ T′ ↠ T′/T′_{1}

By the Lemma there is a morphism α : T′ → T, which makes the diagram commute. Analogously we find β : T → T′. Then α ∘ β is not nilpotent, thus is an automorphism of T. But since also T′ was assumed indecomposable, we find that α is an isomorphism.

Remark 5.11. For the multiplicity of Δ(F) as a subquotient in a Δflag of M ∈ O we get by the above Lemma the formula [M : Δ(F)] = dim_{k} Hom_{O}(M, ∇(F)). In particular this multiplicity does not depend on the choice of a Δ-flag.

Finally we now obtain the looked-for character formula for tilting modules.

Theorem 5.12. Let γ : 𝔤_{0} → k be a semi-infinite character for 𝔤. Then for all E, F ∈ Λ we have

[T (E) : Δ(F)] = [∇(k_{−γ} ⊗ F^{∗}) : L(k_{−γ} ⊗ E^{∗})].

Proof. If 𝔥 ⊂ 𝔤_{0} is a maximal torus, our M_{𝔥} of remark 2.2 (2) is precisely the category O^{Δ} of all objects of O with a finite Δ-flag. Our equivalence O^{Δ} →^{∼} (O^{Δ})^{opp} from remark 2.4 (3) has to transform P_{≤−n}(k_{−γ} ⊗ E^{∗}) into T_{≥n}(E) for all n ≤ |E|, by the very definition of T_{≥n}. Thus for n ≤ |F | we get

[T (E) : Δ(F)] = [T_{≥n}(E) : Δ(F)]
= [P_{≤−n}(k_{−γ} ⊗ E^{∗}) : Δ(k_{−γ} ⊗ F^{∗})]
= [∇(k_{−γ} ⊗ F^{∗}) : L(k_{−γ} ⊗ E^{∗}],

the last equality by the reciprocity formula.

6. Projective objects and tilting modules in categories without grading

For the applications which are the goal of this work, it is convenient to hide the ℤ-grading on the modules. Let in this section as in the two preceding ones 𝔤 = ⨁_{i∈ℤ} 𝔤_{i} be a ℤ-graded Lie-algebra an algebraically closed field over k of characteristic zero such that dim_{k} 𝔤_{i} < ∞ for all i, that 𝔤_{0} is reductive, and that 𝔤 is semisimple for ad𝔤_{0}. But we ask in addition that there is an element ∂ ∈ 𝔤_{0} such that [∂, X] = iX ∀i ∈ ℤ, X ∈ 𝔤_{i}.

As before put 𝔟 = 𝔤_{≥0} and B = U(𝔟). Let Ō denote the category of all 𝔤-modules which are locally finite for 𝔟 and semisimple for 𝔤_{0}. I want to explain briefly how results for Ō can be deduced from the analogous results for O. First by assumption ∂ lies in the center of 𝔤_{0}, thus every M ∈ Ō decomposes under ∂ into eigenspaces M = ⨁_{a∈k} M^{a}. If we consider for every a ∈ k the category

O_{a} = {M ∈ O | M^{a+i} = M_{i} ∀i ∈ ℤ},

we certainly have O = ∏_{a∈k} O_{a}. On the other hand we have also for ā ∈ k/ℤ the subcategory

Ōā = {M ∈ Ō | M^{b} ≠ 0 ⇒ b ∈ ā}

and analogously Ō = ∏_{ā∈k/ℤ} Ōā. But clearly forgetting the ℤ-grading gives us equivalences O_{a} →^{∼} Ōā, which we can use to transfer our results from O to Ō.

To formulate these results for Ō, we need a bit of notation. Let Λ̄ denote the set of all isomorphism classes of finite dimensional irreducible representations of 𝔤_{0}. For E ∈ Λ̄ we consider in Ō the Verma modul Δ(E) = U ⊗_{B} E. It has a unique simple quotient L(E). In addition we consider in Ō for every E ∈ Λ̄ the object ∇(E) = Hom_{U(𝔤_{≤0})}(U, E)^{𝔤_{0}−fin}, that is the space of all 𝔤_{0}-finite vectors in said Hom-space, and L(E) is the socle of ∇(E). On Λ̄ let ≤ be the smallest partial order such that [∇(E) : L(F)] ≠ 0 ⇒ E ≥ F and [Δ(E) : L(F)] ≠ 0 ⇒ E ≥ F. Let ∼ denote the equivalence relation on Λ̄ generated by this partial order. For θ̄ ∈ Λ̄/∼ we put Ōθ̄ = {M ∈ Ō | [M : L(E)] ≠ 0 ⇒ E ∈ θ̄}.

Our theorems of the prededing sections translate into the following.

Theorem 6.1. The functor (Mθ̄)θ̄ ↦ ⨁θ̄ Mθ̄ gives an equivalence of categories

∏_{θ̄∈Λ̄/∼} Ōθ̄ →^{∼} Ō.

Remark 6.2. This generalizes results of [DGK82] and [RCW82].

Definition 6.3. A tilting module with parameter E ∈ Λ̄ is an indecomposable object T = T (E) ∈ Ō such that

1. T admits a Δ-flag starting with T_{1} ≅ Δ(E).

2. Ext1 ; Ō(Δ(F), T) = 0 ∀F ∈ Λ̄.

Theorem 6.4. For every E ∈ Λ̄ there exists in Ō a tilting module T (E) with parameter E. It is unique up to isomorphism.

Remarks 6.5. 1. [Pol91] Remark 5.9 shows that there are enough injectives in Ō, that an injective hull I(E) of L(E) admits a ∇- flag, and that the multiplicities in such a ∇-flag are given by the reciprocity formula

[I(E) : ∇(F)] = [Δ(F) : L(E)] ∀E, F ∈ Λ̄.

2. In addition remark 4.3 shows that L(E) admits a projective cover in Ō if there are but finitely many F ∈ Λ̄ such that F ≥ E. For this projective cover we then have analogously [P (E) : Δ(F)] = [∇(F) : L(E)] ∀E, F ∈ Λ̄.

Let from now on γ : 𝔤_{0} → k be a semi-infinite character of 𝔤, so in particular 𝔤 is generated by 𝔤_{1}, 𝔤_{0} and 𝔤_{−1}. Let Ō^{Δ} denote the category of all objects in Ō with a (finite) Δ-flag.

Theorem 6.6. There is an equivalence of categories Ō^{Δ} → (Ō^{Δ})^{opp} such that short exact sequences correspond to short exact sequences, and such that Δ(E) gets transformed into Δ(k_{−γ} ⊗ E^{∗}), for all E ∈ Λ̄.

If L(E) in Ō admits a projective cover P (E), then P (E) gets transformed into T (k_{−γ} ⊗ E^{∗}) under such an equivalence. But in any case we have

Theorem 6.7. The character of our tilting modules is given by

[T (E) : Δ(F)] = [∇(k_{−γ} ⊗ F^{∗}) : L(k_{−γ} ⊗ E^{∗})] ∀E, F ∈ Λ̄.

7. The case of Kac-Moody algebras

Let 𝔤 be a Kac-Moody algebra, 𝔥 ⊂ 𝔤 its Cartan subalgebra, Π ⊂ 𝔥^{∗} the simple roots. For α ∈ Π the weight spaces 𝔤_{α} and 𝔤_{−α} in 𝔤 generate a subalgebra isomorphic to 𝔰𝔩(2, ℂ) and we let α^{∨} ∈ [𝔤_{α}, 𝔤_{−α}] ⊂ 𝔥 be the vector characterized by ⟨α, α^{∨}⟩ = 2.

Let us consider first on 𝔤 the ℤ-grading with 𝔤_{0} = 𝔥 and 𝔤_{1} = ⨁_{α∈Π} 𝔤_{α}. Since the simple roots are linearly independent, there exists ∂ ∈ 𝔥 such that ⟨α, ∂⟩ = 1 ∀α ∈ Π, thus [∂, X] = iX ∀X ∈ 𝔤_{i}, i ∈ ℤ.

Lemma 7.1. [Ark96] Let ρ ∈ 𝔥^{∗} be such that ⟨ρ, α^{∨}⟩ = 1 ∀α ∈ Π. Then 2ρ is a semi-infinite character for 𝔤.

Proof. We have to show 2ρ([X, Y]) = tr(adXadY : 𝔥 → 𝔥) for all X ∈ 𝔤_{α}, Y ∈ 𝔤_{−β} and α, β ∈ Π. If α ≠ β both sides vanish, if α = β the equality follows from the definitons.

In our case certainly Λ̄ = 𝔥^{∗} and [Δ(λ) : L(μ)] = [∇(λ) : L(μ)] ∀λ, μ ∈ 𝔥^{∗}. For α ∈ Π let s_{α} : 𝔥^{∗} → 𝔥^{∗} denote the involution s_{α}(λ) = λ − ⟨λ, α^{∨}⟩α. The subgroup W ⊂ Aut𝔥^{∗} generated by the s_{α} is called the Weyl group. If we put S = {s_{α} | α ∈ Π}, then (W, S) is a Coxeter system. We define the dot-actions of W on 𝔥^{∗} by the formula x · λ = x(λ + ρ) − ρ. This action does not depend on the choice of ρ. In [Kas90] Kashiwara establishes a conjecture of Deodhar, Gabber and Kac [DGK82] to the effect, that the Kazhdan-Lusztig-polynomials P_{x,y} for (W, S) give Jordan-H”older multiplicities for 𝔤.

Theorem 7.2. [Kas90] Let 𝔤 be symmetrizable. Let λ ∈ 𝔥^{∗} be such that ⟨λ + ρ, α^{∨}⟩ ∈ {1, 2, … } ∀α ∈ Π. Then [Δ(x · λ) : L(ν)] = 0 for ν ∉ W · λ and

[Δ(x · λ) : L(y · λ)] = P_{x,y}(1) ∀x, y ∈ W.

Now theorem 6.7 gives easily the generalization of the main results of [CI89] to symmetrizable Kac-Moody algebras.

Corollary 7.3. Suppose 𝔤 symmetrizable. Let μ ∈ 𝔥^{∗} be such that ⟨μ + ρ, α^{∨}⟩ ∈ {−1, −2, … } ∀α ∈ Π. Then [T (y · μ) : Δ(ν)] = 0 for ν ∉ W · μ and

[T (y · μ) : Δ(x · μ)] = P_{x,y}(1) ∀x, y ∈ W.

Let us consider also the parabolic case. Let Π_{f} ⊂ Π be a set of simple roots such that the 𝔤_{α} with ±α ∈ Π_{f} generate a finite dimensional (necessarily semisimple) subalgebra of 𝔤. Then 𝔤 also admits a ℤgrading such that 𝔥 ⊂ 𝔤_{0}, 𝔤_{α} ⊂ 𝔤_{0} if ±α ∈ Π_{f}, and 𝔤_{α} ⊂ 𝔤_{1} if α ∈ Π − Π_{f}, and for this ℤ-grading our conditions from the beginning of section 6 are satisfied as well. Let ρ_{f} ∈ 𝔥^{∗} denote the half sum of positive roots of 𝔤_{0}.

Lemma 7.4. There is a character γ = γ_{f} : 𝔤_{0} → ℂ which coincides on 𝔥 with 2ρ − 2ρ_{f}. Every such character γ is a semi-infinite character for 𝔤.

Proof. Certainly 𝔤 is generated by 𝔤_{1}, 𝔤_{0} and 𝔤_{−1}. for our new grading as well. We have ⟨ρ, β^{∨}⟩ = ⟨ρ_{f}, β^{∨}⟩ = 1 ∀β ∈ Π_{f}, thus 2ρ − 2ρ_{f} disappears on [𝔤_{0}, 𝔤_{0}] ∩ 𝔥 and can indeed be extended to a character γ of 𝔤_{0}. The only thing left to show is the formula

γ([X, Y]) = tr(adXadY : 𝔤_{0} → 𝔤_{0}) ∀X ∈ 𝔤_{1}, Y ∈ 𝔤_{−1}.

If this holds for fixed X ∈ 𝔤_{1} with arbitrary Y ∈ 𝔤_{−1}, then it also holds for [A, X] with arbitrary Y ∈ 𝔤_{−1} and A ∈ 𝔤_{0}. We leave this verification to the reader and then only have to check that

γ([X_{α}, X_{β}]) = tr(adX_{α}adX_{β} : 𝔤_{0} → 𝔤_{0})

for α ∈ Π − Π_{f}, β a root of 𝔤 with 𝔤_{β} ⊂ 𝔤_{−1}, and X_{β} ∈ 𝔤_{β}. If β ≠ −α both sides of our equation vanish, and we are left with the case β = −α. We then put X_{β} = Y_{α} and decompose 𝔤_{0} = 𝔫_{0}^{+} ⊕ 𝔥 ⊕ 𝔫_{0}^{−} under the adjoint action of 𝔥. All three summands are stable under adX_{α}adY_{α}. Furthermore [Y_{α}, 𝔫_{0}^{+}] = 0 = [X_{α}, 𝔫_{0}^{−}], since the weight of these brackets doesn’t belong to the root system of 𝔤. Without restriction suppose [X_{α}, Y_{α}] = α^{∨}. We get

tr(adX_{α}adY_{α} : 𝔤_{0} → 𝔤_{0}) = tr(adX_{α}adY_{α} : 𝔥 → 𝔥)
+tr(ad(α^{∨}) : 𝔫_{0}^{−} → 𝔫_{0}^{−})
= 2 − 2ρ_{f}(α^{∨})
= ⟨2ρ − 2ρ_{f}, α^{∨}⟩.

In the parabolic case we can identify Λ̄ with the set

𝔥_{f}^{∗} = {λ ∈ 𝔥^{∗} | ⟨λ, α^{∨}⟩ ∈ ℕ ∀α ∈ Π_{f}},

by associating to E ∈ Λ̄ its highest weight λ(E) ∈ 𝔥^{∗}. For λ = λ(E) ∈ 𝔥_{f}^{∗} we also write Δ^{f}(λ) instead of Δ(E) and call this object the parabolic Verma module of highest weight λ. Analogously we define ∇^{f}(λ), T^{f}(λ) and L^{f}(λ). However we will never use L^{f}(λ) since clearly L^{f}(λ) = L(λ).

Certainly we have [∇^{f}(λ) : L(μ)] = [Δ^{f}(λ) : L(μ)] ∀λ, μ ∈ 𝔥_{f}^{∗}. To calculate these multiplicities we take λ = λ(E) ∈ 𝔥_{f}^{∗} and consider for the 𝔤_{0}-module E following [BGG75] the resolution

0 → M_{r} → … → M_{1} → M_{0} → E → 0.

So here M_{i} = ⨁_{l(z)=i} Δ_{f}(z · λ) where Δ_{f}(μ) denotes the Verma for 𝔤_{0} with heighest weight μ and the sum runs over all elements z of length i of the Weyl group W_{f} = ⟨s_{α} | α ∈ Π_{f}⟩ of 𝔤_{0}. (Remark that z(λ + ρ_{f}) − ρ_{f} = z(λ + ρ) − ρ ∀z ∈ W_{f}, since ⟨ρ, α^{∨}⟩ = 1 = ⟨ρ_{f}, α^{∨}⟩ ∀α ∈ Π_{f}.)

Applying the functor U⊗_{B} to this exact sequence, we get an exact sequence

0 → U ⊗_{B} M_{r} → … → U ⊗_{B} M_{0} → Δ^{f}(λ) → 0

with U ⊗_{B} M_{i} = ⨁_{l(z)=i} Δ(z · λ). This implies

[Δ^{f}(λ) : L(ν)] = ∑_{z∈W_{f}}(−1)^{l(z)}[Δ(z · λ) : L(ν)]

for all λ ∈ 𝔥_{f}^{∗}, ν ∈ 𝔥^{∗}. (In this argument the BGG-resolution can be replaced by the Weyl character formula, if one takes the time to introduce suitable Grothendieck groups.) Let now W^{f} ⊂ W denote the set of shortest representatives for the right cosets of W_{f}. Thus multiplication gives a bijection W_{f} × W^{f} →^{∼} W. If ℕΠ denotes the set of all linear combinations of simple roots with integral nonnegative coefficients, we can describe W^{f} alternatively as

W^{f} = {x ∈ W | x^{−1}Π_{f} ⊂ ℕΠ}.

In particular for λ ∈ 𝔥^{∗} with ⟨λ, α^{∨}⟩ ∈ ℕ ∀α ∈ Π we have x · λ ∈ 𝔥_{f}^{∗} ∀x ∈ W^{f}.

For x, y ∈ W^{f} let n_{x,y} ∈ ℤ[v] denote one of the two corresponding parabolic Kazhdan-Lusztig polynomials in [Deo87], in the normalization and notation of [Soe97].

Proposition 7.5. Let 𝔤 be symmetrizable. Let λ ∈ 𝔥^{∗} be such that ⟨λ, α^{∨}⟩ ∈ ℕ ∀α ∈ Π. Then [Δ^{f}(x · λ) : L(ν)] = 0 for ν ∉ W^{f} · λ and

[Δ^{f}(x · λ) : L(y · λ)] = n_{x,y}(1) ∀x, y ∈ W^{f}.

Proof. By Deodhar [Deo87, Soe97] our parabolic polynomials satisfy n_{x,y}(1) = ∑_{z∈W_{f}} (−1)^{l(z)}P_{zx,y}(1).

This gives a character formula for certain parabolic tilting modules T^{f}(λ). More precisely, let w_{f} ∈ W_{f} be the longest element. We define a new action of W on 𝔥^{∗} by x ▪ λ = (w_{f}xw_{f}) · λ.

Corollary 7.6. Let 𝔤 be symmetrizable. Suppose μ ∈ 𝔥^{∗} is such that ⟨−w_{f}μ − 2ρ + 2ρ_{f}, α^{∨}⟩ ∈ ℕ ∀α ∈ Π. Then [T^{f}(y ▪ μ) : Δ^{f}(ν)] = 0 for ν ∉ W^{f} ▪ μ and

[T^{f}(y ▪ μ) : Δ^{f}(x ▪ μ)] = n_{x,y}(1) ∀x, y ∈ W^{f}.

Proof. We only have to calculate. First remark ρ − ρ_{f} = w_{f}(ρ − ρ_{f}) = w_{f}ρ + ρ_{f}, thus w_{f}ρ = ρ − 2ρ_{f}. Furthermore −w_{f}λ(E) is the highest weight of E^{∗}. Let γ = 2ρ − 2ρ_{f} be as above. If x ▪ μ is the highest weight of E, then the highest weight of k_{−γ} ⊗ E^{∗} is precisely x · λ for λ = −w_{f}μ − 2ρ + 2ρ_{f}. We leave the verification to the reader. The Corollary follows, in particular we get y ▪ μ ∈ 𝔥_{f}^{∗} ∀y ∈ W^{f}.

8. Tranlation into results for quantum groups

For the application to quantum groups at roots of unity the important case is to take 𝔤 the affinization of a simple complex Lie algebra 𝔤̀ and Π_{f} the set of all simple roots except the “affine root”. To introduce the necessary notation I will start with some recollections concerning the affinization of a simple complex Lie algebra 𝔤̀.

One starts with the loop algebra L𝔤̀ = 𝔤̀ ⊗_{ℂ} ℂ[t, t^{−1}]. Its elements will be understood as maps of the circle S^{1} to 𝔤̀ and written

X = X(t) = ∑ X_{i}t^{i}.

Let κ̀ : 𝔤̀×𝔤̀ → ℂ be the Killing form. We consider the central extension 𝔤̃ = L𝔤̀ ⊕ ℂz of L𝔤̀ with bracket

[X + aK, Y + bK] = [X, Y] − res(κ̀(X, dY))z

for all X, Y ∈ L𝔤̀ and a, b ∈ ℂ, where we understand d(∑ Y_{i}t^{i}) = ∑ iY_{i}t^{i−1}dt and res(∑ a_{i}t^{i}dt) = a_{−1}. In fact we should not prefer the Killing form in this construction, but rather interpret ℂz as the dual of the one-dimensional space of all 𝔤̀-invariant bilinear forms on 𝔤̀.

Finally we extend the derivation ∂ = t\frac{∂}{∂t} of L𝔤̀ by the rule ∂(z) = 0 to a derivation of 𝔤̃, and adjoining this derivation to 𝔤̃ we get the so-called affinization 𝔤 = 𝔤̃ ⊕ ℂ∂ of 𝔤̀ with bracket

[X + a∂, Y + b∂] = [X, Y] + a∂(Y) − b∂(X)

for all X, Y ∈ 𝔤̃ and a, b ∈ ℂ.

The center 𝔷 of 𝔤 is the line ℂz. We equip 𝔤 with the ℤ-grading 𝔤_{0} = 𝔤̀ ⊕ ℂz ⊕ ℂ∂, 𝔤_{i} = 𝔤 ⊗ t^{i} if i ≠ 0. Thus 𝔤_{i} is the eigenspace of ad∂ with eigenvalue i, and the semi-infinite characters γ : 𝔤_{0} → ℂ are precisely the linear maps γ : 𝔤_{0} → ℂ such that γ(𝔤̀) = 0 and γ(z) = 1. The Lie algebra 𝔤 is an affine Kac-Moody algebra, and the grading corresponds to a choice Π_{f} ⊂ Π such that Π − Π_{f} consists of a single element α_{0}.

In their articles [KL93, KL94] Kazhdan und Lusztig consider for c ∈ ℂ a certain subcategory given by suitable finitness conditions in the category Õ(z = c) of all 𝔤̃-modules, that are killed by (z − c) and on which 𝔤̃_{≥0} acts locally finitely and 𝔤̃_{>0} locally nilpotently. We want to translate our results for Ō into results for Õ(z = c) and need results of [Kac90].

For every M ∈ Ō the Casimir Ω gives a locally finite endomorphism Ω = Ω_{M} ∈ End_{𝔤}M, and for every morphism g : M → N we have g ∘ Ω_{M} = Ω_{N} ∘ g. For a ∈ ℂ let Ō(Ω ≃ a) denote the subcategory of all objects of Ō, on which (Ω − a) acts locally nilpotently.

Proposition 8.1. [Pol91] Let c ∈ ℂ, c ≠ −1/2. Then forgetting the action of ∂ gives an equivalence of categories

Ō(Ω ≃ 0, z = c) →^{∼} Õ(z = c).

Remarks 8.2. 1. The case z = −1/2 is precisely the “critical level”. Instead of z Kac uses the “canonical” central element K = 2hz with h = h^{∨} the Coxeter number.

2. Under the assumptions of the Proposition forgetting the action of ∂ gives more generally equivalences

Ō(Ω ≃ a, z = c) →^{∼} Õ(z = c)

for all a ∈ ℂ. Indeed the eigenvalue of Ω can be adjusted to every given number if we change the action of ∂ by an additive constant, see [Kac90], 12.8.3.

Proof. Given a representation M of 𝔤̃ such that every m ∈ M is killed by 𝔤̀⊗t^{i} for i ≫ 0, one defines as in [Kac90], 12.8.4 the zeroth Sugawara operator T_{0} : M → M. If z acts by a scalar c ≠ −1/2 on M, the formula ∂ = −(2h(2c + 1))^{−1}T_{0} then determines a (functorial) extension of the action of 𝔤̃ on M to an action of 𝔤, see[Kac90], Corollary 12.8 (a). In case M already was a representation of 𝔤, we have by [Kac90], 12.8.5 the relation

T_{0} = −2h(2c + 1)∂ + Ω.

Now we define an inverse to the functor of the Proposition. Certainly T_{0} acts locally finitely on M ∈ Õ(z = c). So if we extend our 𝔤̃-action on M by ∂ = −(2h(2c + 1))^{−1}T_{0} to a 𝔤-action, ∂ acts locally finitely and Ω acts as zero. If we now define a new action of ∂ on M as the semisimplification of the first action, we obtain a new action of 𝔤 on M, and one sees easily that this gives a functor

Õ(z = c) → Ō(Ω ≃ 0, z = c).

On the other hand the action of ∂ on an object of Ō(Ω ≃ 0, z = c) has to be the semisimplification of the action of −(2h(2c + 1))^{−1}T_{0}, by the relations between T_{0}, ∂ and Ω recalled above. Thus we really found an inverse to the functor of the Proposition.

Suppose from now on that 𝔤̀ has type ADE. To simplify we work with the canonical central element K = 2hz, thus the critical level is K = −h. In this case Kazhdan and Lusztig [KL93, KL94] show for many l ∈ ℂ − ℚ_{≤0} an equivalence of categories

Õ^{e}(K = −h − l) ≅ U_{ζ}-mod^{e,1}.

Here Õ^{e} denotes the category of all finite length objects in Õ, we put ζ = exp(−πi/l), and U_{ζ}-mod^{e,1} means the category of all finite dimensional representations of type 1 of the quantum group with divided powers U_{ζ}. In particular for l ∈ ℤ_{≥h} the block B ⊂ U_{ζ}-mod^{e,1} of the trivial representation of U_{ζ} is equivalent to the block of Ō containing L(μ) with μ = −((h + l)/2h)γ. (This holds for every choice of a semi-infinite character γ = 2ρ − 2ρ_{f}.)

To prove conjecture 7.2 from [Soe97] concerning the character of tilting modules for quantum groups at roots of unity l > 33, we thus have to check in Ō the formula

[T^{f}(y ▪ μ) : Δ^{f}(x ▪ μ)] = n_{x,y}(1)

for all x, y ∈ W^{f}. By Corollary 7.6 it is sufficient to check ⟨−w_{f}μ − 2ρ + 2ρ_{f}, α^{∨}⟩ ∈ ℕ for all α ∈ Π.

Now a possible definition of the Coxeter number is ⟨ρ_{f}, α_{0}^{∨}⟩ = −h+1, we thus get ⟨γ, α_{0}^{∨}⟩ = 2h. Furthermore we have ⟨γ, α^{∨}⟩ = 0 ∀α ∈ Π_{f}. Since w_{f}γ = γ we get −w_{f}μ − 2ρ + 2ρ_{f} = ((h + l)/2h)γ − γ = ((l − h)/2h)γ, thus this weight vanishes on all coroots α^{∨} for α ∈ Π_{f} and takes on α_{0}^{∨} the value l − h ∈ ℕ.

If 𝔤̀ is not of type ADE, the argument is more or less the same. The only problem is that the Kazhdan-Lusztig-conjectures for affine Lie algebras in positive level are only known for integral weights. Here there are still gaps in the literature to be filled.

How one can get characters for tilting modules “on the walls” from the character formulas in the principal block is explained in [Soe97].

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Universität Freiburg, Mathematisches Institut, Eckerstraße 1, D-79104 Freiburg, Germany

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