CHARACTER FORMULAS FOR TILTING MODULES OVER QUANTUM GROUPS AT ROOTS OF ONE

WOLFGANG SOERGEL

1. Motivation

Let k be an algebraically closed field, n ≥ 1 an integer. Our motivating problem is the study of the irreducible representations of the symmetric group S_{n} over k, in other words of the irreducible modules over the group ring kS_{n}. In case chark = 0 their classification is well known, we have a bijection

IrrkS_{n} ↔ {Partitions of n}.

In case chark = p > 0 the classification is less well known, we have a bijection

IrrkS_{n} ↔ \{ \begin{matrix}Partitions of n which do not \\ have p or more equal pieces \end{matrix} \} .

The next question is, given such a partition, to determine the dimension of the corresponding irreducible representation of the symmetric group. In case chark = 0, these dimensions are again well known. We just have to display our partition as a Young diagram and count the standard tableaux of this shape, i.e. the dimension in question is the number of possibilities we have to enumerate the boxes of our Young diagram such that each row and column is increasing. For all this see [Jam78]. In case chark = p > 0, the dimensions are not even known for all partitions of n into three pieces.

This is the problem where tilting modules suggest a new line of attack [Don93, Erd94, Erd95, Mat96]. Namely consider a finite dimensional vector space V over k of dimension dim_{k} V ≥ n. Then “Schur-Weyl duality” (see [CL74] or 2.9 below for the case of arbitrary characteristic) says that

End_{GL(V)}V^{⊗n} = kS_{n}

with the symmetric group permuting the factors of a tensor in V^{⊗n}.

If chark = 0, we know that V^{⊗n} decomposes into a direct sum of irreducible subrepresentations under GL(V). In general, this is no longer true. But suppose

V^{⊗n} ≅ ⨁_{λ∈Λ} T (λ)^{m(λ)}

is a decomposition of the GL(V)-module V^{⊗n} into indecomposable (not in general irreducible) direct summands, where the isomorphism classes of indecomposables T (λ) appearing are parametrized by a suitable set Λ and occur with multiplicity m(λ) > 0. Then it is clear from abstract algebra that the quotient of kS_{n} by its Jacobson radical is just a product of matrix algebras of the form

kS_{n}/radkS_{n} ≅ Π_{λ∈Λ}M(m(λ), k)

and we get a bijection

IrrkS_{n}  ⟷ Λ
( \begin{matrix}some irreducible \\ of dimension m(λ) \end{matrix} ) ↦ λ.

Therefore it is of the utmost importance to determine the decomposition of V^{⊗n} into indecomposables. In the next section we will following [Don93] define what is a “tilting module” for G = GL(V) or more generally for an arbitrary reductive algebraic group G. We will see that the indecomposable tilting modules are classified by their highest weights, and that all the summands T (λ) of V^{⊗n} above are actually tilting modules.

Since the character of V^{⊗n} is known, we just need to determine the character of the indecomposable tilting module for any given highest weight to obtain the looked-for decomposition of V^{⊗n}. We can only solve the analogous problem with “algebraic groups in positive characteristic” replaced by “quantum groups at roots of one”, and conjecture (see [And97]) that if the highest weight is not too big, the quantized problem has the same solution as the original one. In the hope that the reader is now sufficiently motivated let me go on to define tilting modules.

2. Tilting modules for reductive algebraic groups

Let k be an algebraically closed field, G ⊃ B ⊃ H a connected reductive algebraic group over k, a Borel subroup and a maximal torus (see [Jan87] for all this foundational material). For example one can consider G = GL(n, k), take for B all upper triangular matrices and for H all diagonal matrices. The inclusion H ⊂ B admits a unique left inverse B ↠ H which allows us to consider every character of H as a character of B. For λ in the character lattice X = X(H) of H we consider the induced representation

∇(λ) = ind_{B}^{G}k_{λ}
= \{f : G → k | \begin{matrix}f is algebraic and satisfies \\ f(xb) = λ(b)^{−1}f(x) ∀x ∈ G, b ∈ B \end{matrix} \}

with the action of g ∈ G on f ∈ ∇(λ) defined by (gf)(x) = f(g^{−1}x). If we let R ⊂ X be the root system of G, take as positive roots the complement R^{+} = R − R(B) of the roots of B, and let X^{+} ⊂ X be the corresponding dominant weights, then we have ∇(λ) ≠ 0 iff λ ∈ X^{+}.

In case chark = 0 our ∇(λ) is precisely the simple representation of G with highest weight λ. In general, the character of ∇(λ) is still given by the Weyl character formula, but the ∇(λ) are no longer simple for all λ, as we can readily see from the example G = GL(2, k). In this case, letting ρ be half the positive root, we can identify the representation ∇(nρ) with the obvious representation k[X, Y]^{n} of GL(2, k) on polynomials in two variables homogeneous of degree n, and for chark = p obviously kX^{p} + kY^{p} ⊂ k[X, Y]^{p} is an invariant subspace. We can further see that this subspace has no invariant complement, so ∇(pρ) is not completely reducible if chark = p > 0.

Definition 2.1. [Don93] A rational representation T of G is called a “tilting module” if and only if both T and its dual T^{∗} admit a finite ∇-flag, i.e. a finite filtration by G-stable subspaces such that all sucessive subquotients are isomorphic to some ∇(λ). In particular a tilting module by definition is always of finite dimension over k.

Remark 2.2. In the terminology of Donkin (due to Ringel [Rin91]) such representations would in fact be called a “partial tilting modules”. However common usage is now to just call them tilting modules.

Examples 2.3. The trivial representation k = ∇(0) is always tilting. If more generally ∇(λ) is simple, one can show that ∇(λ)^{∗} ≅ ∇(−w_{∘}λ) for w_{∘} the longest element of the Weyl group, hence every simple ∇(λ) is tilting. As a special case of this we see that V is a tilting module for GL(V).

To have a non-trivial example, let us consider the tensor product ∇(ρ) ⊗ ∇(pρ − ρ) for G = GL(2, k), where chark = p > 0. This is certainly selfdual and one may see by hand that it fits into a short exact sequence

∇(pρ − 2ρ) ↪ ∇(ρ) ⊗ ∇(pρ − ρ) ↠ ∇(pρ).

Therefore ∇(ρ) ⊗ ∇(pρ − ρ) is a tilting module for G = GL(2, k).

The following theorem collects some of the reasons, why tilting modules are interesting.

Theorem 2.4. 1. Every summand of a tilting module is tilting. 2. The tensor product of two tilting modules is tilting.

3. The indecomposable tilting modules are classified by their highest weights, more precisely we have a bijection

\{ \begin{matrix}indecomposable tilting modules, \\ up to isomorphism \end{matrix} \} ↔  X^{+}
T  ↦ “the highest weight of T ”

Notation 2.5. We denote by T (λ) the indecomposable tilting module with highest weight λ.

Proof. Let us give some indications on where this theorem comes from. Part (2) follows easily from the following theorem due in full generality to [Mat90]:

Theorem 2.6. For all λ, μ ∈ X^{+} the tensor product ∇(λ) ⊗ ∇(μ) admits a ∇-flag.

To explain part (1), let us define Δ(λ) = ∇(−w_{∘}λ)^{∗} for every λ ∈ X. (The parametrization of the Δ(λ) is just set up in such a way that Δ(λ) and ∇(λ) have the same character.) Let Ext_{G} denote extensions in the category of all rational representations of G. Part (1) then follows easily from the following result of Donkin [Don81], (see also [Jan87], II, 4.16):

Proposition 2.7. For a finite dimensional rational representation T of G the following are equivalent:

1. T admits a ∇-flag.

2. Ext_{G}^{1}(Δ(λ), T) = 0 ∀λ ∈ X^{+}.

Part (3) of the theorem can be proved in a very general context, as is explained in the next section following [Rin91]. For this we use the usual partial order on X^{+} given by λ ≥ μ iff λ ∈ μ + ℕR^{+} and then only need to know that Hom_{G}(Δ(λ), Δ(μ)) = 0 unless λ ≤ μ, Ext_{G}^{1}(Δ(λ), Δ(μ)) vanishes unless λ < μ, and both are finite dimensional for all λ, μ. This can be deduced from results in [Jan87] without too much difficulty.

Remark 2.8. From the theorem and the fact that V is tilting for GL(V) it is clear that all direct summands of V^{⊗n} are tilting, too. (By the way we only need for this 2.6 in the easy case ∇(μ) = V.) In any case it is clear now that a Krull-Schmid-decomposition of the GL(V)-module V^{⊗n} into indecomposable direct summands has to be of the form

V^{⊗n} = ⨁_{λ∈X^{+}} T (λ)^{m(λ)}

where the m(λ) are suitable multiplicities m(λ) ≥ 0 and the T (λ) are our indecomposable tilting modules from 2.5. Already for G = GL(3, k) however the characters of the T (λ) are not all known, and what’s worse, there isn’t even a general conjecture. If λ is in the “lowest p^{2}-alcove”, then T (λ) is conjectured to have the same character as its quantum analogue, see [And97].

Remark 2.9. Let me explain how the above results imply the fundamental formula End_{GL(V)}V^{⊗n} = kS_{n} (if k is infinite and dim_{k} V ≥ n). There is no problem in verifying explicitely that End_{kS_{n}}V^{⊗n} is the k-linear subspace of End_{k}V^{⊗n} generated by the image of GL(V). If chark = 0, then V^{⊗n} is a semisimple kS_{n}-module and thus by the Jacobson density theorem kS_{n} surjects onto the bicommutant of its image in End_{k}V^{⊗n}, i.e. we have a surjection kS_{n}→ →End_{GL(V)}V^{⊗n}.

For chark is arbitrary it is still clear that kS_{n} → End_{GL(V)}V^{⊗n} is an injection in case dim_{k} V ≥ n. The problem is surjectivity. For this we will count dimensions and just have to show that the dimension of End_{GL(V)}V^{⊗n} is independent of the ground field k. For this in turn we need to know that dim_{k} Hom_{G}(Δ(λ), ∇(μ)) = δ_{λ,μ} and Ext_{G}^{1}(Δ(λ), ∇(μ)) = 0 for all λ, μ ∈ X^{+}. (The first statement can be found in say [Jan87], the second one also is a special case of 2.7 above.) Then if T has a finite Δ-flag where Δ(λ) occurs with multiplicity (T : Δ(λ)) and T′ has a finite ∇-flag where ∇(λ) occurs with multiplicity (T′ : ∇(λ)), we get

dim_{k} Hom_{G}(T, T′) = ∑_{λ}(T : Δ(λ))(T′ : ∇(λ)).

In particular, we get

dim_{k} End_{GL(V)}V^{⊗n} = ∑_{λ}(V^{⊗n} : Δ(λ))(V^{⊗n} : ∇(λ))

which is indeed independent from the ground field k.

3. Existence and unicity of tilting modules in an abstract context

Suppose k is a field, C an abelian k-category, and {Δ(λ)}_{λ∈Λ} a collection of indecomposable objects of C parametrized by a partially ordered set (Λ, ≤), such that the following conditions are satisfied:

1. Hom(Δ(λ), Δ(μ)) = 0 unless λ ≤ μ.

2. Ext_{C}^{1}(Δ(λ), Δ(μ)) = 0 unless λ < μ.

3. dim_{k} Hom_{C}(Δ(λ), Δ(μ)) < ∞, dim_{k} Ext_{C}^{1}(Δ(λ), Δ(μ)) < ∞ ∀λ, μ.

4. For every λ ∈ Λ there are only finitely many elements below it. In this situation we have

Proposition 3.1. [Rin91] For every λ ∈ Λ there exists a unique (up to non-unique isomorphism) indecomposable object T = T (λ) ∈ C, which satisfies the following two conditions:

(a) Ext_{C}^{1}(Δ(ν), T) = 0 ∀ν ∈ Λ.

(b) T admits a finite Δ-flag starting with Δ(λ) ⊂ T.

Remark 3.2. By a finite Δ-flag of an object T ∈ C we mean a filtration 0 = T_{0} ⊂ T_{1} ⊂ T_{2} ⊂ … T_{r} = T such that T_{i}/T_{i−1} ≅ Δ(λ_{i}) for suitable λ_{i} ∈ Λ.

Definition 3.3. T (λ) might be called the indecomposable tilting module with parameter λ, but in this generality the terminology is not commonly used.

Proof. Existence: We prove the existence of T (λ) by induction on the number of elements below λ. If λ is already minimal, we can take T (λ) = Δ(λ). If not, we find a minimal μ ∈ Λ with μ < λ. By induction, we know there exists T̃ ∈ C such that Ext_{C}^{1}(Δ(ν), T̃) = 0 for all ν ≠ μ and that T̃ admits a finite Δ-flag with subquotients Δ(ν), ν ≠ μ. If Ext_{C}^{1}(Δ(μ), T̃) = 0 we can take T̃ = T (λ) and are through. If not, we have at least dim_{k} Ext_{C}^{1}(Δ(μ), T̃) < ∞ by condition (3). Put T̃ = T_{0}, choose a nonzero element e of this Ext-group, and represent it by a short exact sequence

T_{0} ↪ T_{1} ↠ Δ(μ).

A segment of the corresponding long exact sequence of Ext-groups is

Hom_{C}(Δ(μ), Δ(μ)) → Ext_{C}^{1}(Δ(μ), T_{0}) → Ext_{C}^{1}(Δ(μ), T_{1}) → 0,

the last zero since we have Ext_{C}^{1}(Δ(μ), Δ(μ)) = 0 by condition (1). Now the first arrow of these maps id ∈ Hom_{C}(Δ(μ), Δ(μ)) to our nonzero element e. Thus dim_{k} Ext_{C}^{1}(Δ(μ), T_{1}) < dim_{k} Ext_{C}^{1}(Δ(μ), T_{0}) and by other long exact sequences still Ext_{C}^{1}(Δ(ν), T_{1}) = 0 if ν ≠ μ. Continuing this way, we construct inductively T_{2}, T_{3}, … until we arrive at some T_{r} that fits into a short exact sequence

T̃ ↪ T_{r} ↠ Δ(μ)^{r}

and satisfies Ext_{C}^{1}(Δ(ν), T_{r}) = 0 ∀ν ∈ Λ. In case End_{C}Δ(μ) = k it can be shown that T_{r} is actually indecomposable, but in our general situation this need not be true. To circumvent this problem we rather show that if m is the smallest possible integer such that there exists a short exact sequence

T̃ ↪ T ↠ Δ(μ)^{m}

with T satisfying Ext_{C}^{1}(Δ(ν), T) = 0 ∀ν, then this T is indecomposable and hence our looked-for T (λ). Indeed, we can describe T̃ as the biggest subobject of T killed by all morphisms T → Δ(μ), and hence any decomposition T = T′ ⊕ T′′ induces a decomposition T̃ = T̃′ ⊕ T̃′′. Since T̃ is indecomposable, we may assume T̃ = T̃′, T̃′′ = 0 and get T′/T̃ ⊕ T′′ ≅ Δ(μ)^{m}. Now the Krull-Schmid theorem implies T′/T̃ ≅ Δ(μ)^{n} with n ≤ m, but by minimality of m we have necessarily n = m, hence T′′ = 0.

Unicity: Since T is indecomposable, End_{C}T admits no idempotents except zero and one. Since T has a finite Δ-flag, End_{C}T is of finite dimension over k by (2). Together this implies that every element of End_{C}T is either nilpotent or an isomorphism, by the Lemma of Fitting.

Suppose now T, T′ ∈ C are two indecomposables satisfying (a) and (b). Consider the diagram

Δ(λ) ↪ T ↠ coker
||
Δ(λ) ↪ T′ ↠ coker′

with short exact horizontals. By (b) coker has a Δ-flag, thus by (a) we have Ext_{C}^{1}(coker, T′) = 0, thus the restriction Hom_{C}(T, T′) → Hom_{C}(Δ(λ), T′) is a surjection and we can find α : T → T′ inducing the identity on Δ(λ). Similarily we also find β : T′ → T inducing the identity on Δ(λ). Since β ∘ α ∈ End_{C}T is not nilpotent, it has to be an isomorphism, and the same holds for α ∘ β. We conclude T ≅ T′.

The multiplicity of Δ(ν) as subquotient in a Δ-flag of T (λ) will be denoted by (T (λ) : Δ(ν)). This number is independent of the Δ-flag. Indeed, if we enumerate the elements below λ as λ_{1}, …, λ_{n} such that λ_{i} > λ_{j} ⇒ i < j, in particular λ_{i} = λ and λ_{n} is minimal, then by the vanishing of the relevant Ext_{C}^{1} any Δ-flag can be transformed without changing its multiplicities to a Δ-flag where first come the subquotients Δ(λ_{1}), then the Δ(λ_{2}) etc. It is then clear from our inductive construction of T (λ) that all multiplicities are well determined.

4. Character formulas for indecomposable tilting modules over graded Lie algebras

We will explain next how to determine the characters of indecomposable tilting modules for quantum groups at roots of unity or equivalently [KL93, KL94] how to solve the translated problem in a suitable category O of representations of an affine Lie algebra. First we will work with an arbitrary ℤ-graded Lie-algebra 𝔤 = ⨁_{i∈ℤ} 𝔤_{i} over an arbitrary field k subject only to the following three conditions

1. dim_{k} 𝔤_{i} < ∞ ∀i ∈ ℤ.

2. 𝔤 is generated by 𝔤_{−1}, 𝔤_{0}, 𝔤_{1}.

3. There exists a character γ : 𝔤_{0} → k such that tr(adXadY : 𝔤_{0} →

𝔤_{0}) = γ([X, Y]) for all X ∈ 𝔤_{1}, Y ∈ 𝔤_{−1}. Let us put 𝔤_{<0} = 𝔫, 𝔤_{≥0} = 𝔟 and denote the enveloping algebras of 𝔤, 𝔫, 𝔟 by U, N, B. Let M denote the category of all ℤ-graded 𝔤modules, which are graded free of finite rank over N. The following result is due to Archipov [Ark97] and can also be found in [Soe97a].

Theorem 4.1. There exists an equivalence of categories M → M^{opp} such that short exact sequences of 𝔤-modules on both sides correspond, and that U ⊗_{B} E gets mapped to U ⊗_{B} (k_{−γ} ⊗ E^{∗}) for every finite dimensional irreducible ℤ-graded representation E of 𝔤_{0}.

Proof. I will only give some indications on how this result may be proven, more details can be found in [Soe97a]. Denote for a ℤ-graded space M = ⨁ M_{i} by M^{⊛} its graded dual, so that (M^{⊛})_{i} = (M_{−i})^{∗}. We make N^{⊛} into a ℤ-graded N-bimodule in the most obvious way, the left action of N on itself giving rise to the right action on N^{⊛} and vice versa. The key ingredient to our equivalence will be a very peculiar ℤ-graded U-bimodule S, the so-called “semi-regular bimodule” introduced in [Vor93, Ark97], whose existence is assured by the following

Proposition 4.2. There exists a ℤ-graded U-bimodule S = S_{γ} along with an inclusion ι : N^{⊛} ↪ S of ℤ-graded N-bimodules such that the following holds:

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 is compatible with the adjoint action of 𝔤_{0} on both spaces, more precisely ι(f ∘ adH) + (adH)ι(f) = ι(f)γ(H) for all H ∈ 𝔤_{0} and f ∈ N^{⊛}.

This Proposition can be checked by brute force, but I still don’t know a good proof. Anyhow, we can now write down the equivalence of categories claimed by the theorem as the functor M ↦ (S ⊗_{U} M)^{⊛}, and it is not difficult to check that this has the requested properties.

Suppose now we are working over a ground field k of characteristic zero and 𝔤 is semisimple under the adjoint action of 𝔤_{0}. Let O be the category of all ℤ-graded 𝔤-modules which are locally finite over 𝔤_{≥0} and semisimple as modules over 𝔤_{0}. Let Λ be the set of isomorphism classes of finite dimensional simple ℤ-graded 𝔤_{0}-modules. Any E ∈ Λ is necessarily concentrated in just one degree. For E ∈ Λ the Verma module Δ(E) = U ⊗_{B} E admits a unique simple quotient L(E), and in this way Λ parametizes the simple objects of O up to isomorphism. Adapting the arguments of section 3 to our situation, we can prove

Theorem 4.3. For every E ∈ Λ there exists a unique (up to nonunique isomorphism) indecomposable object T = T (E) ∈ O which satisfies the following two conditions:

1. Ext_{O}^{1}(Δ(F), T (E) = 0 ∀F ∈ Λ.

2. T admits a Δ-flag starting with Δ(E), i.e. a filtration 0 = T_{0} ⊂ T_{1} ⊂ T_{2} ⊂ … such that T_{1} ≅ Δ(E), ⋃ T_{i} = T and T_{i}/T_{i−1} ≅ Δ(F_{i}) for suitable F_{i} ∈ Λ.

Certainly we call T (E) the tilting module with parameter E. The main theorem is a formula for the multiplicity (T (E) : Δ(F)) of Δ(F) as subquotient in a Δ-flag of T (E) as above. For this we have to introduce the module

∇(E) = (U ⊗_{U(𝔤_{≤0})} E^{∗})^{⊛},

which is the correct generalization of what is known as the “dual Verma” in the Kac-Moody set-up. Let [∇(E) : L(F)] denote the Jordan-Hölder-multiplicity of L(F) in ∇(E). The “abstract character formula for tilting modules” then says

Theorem 4.4. (T (E) : Δ(F)) = [∇(k_{−γ} ⊗ F^{∗}) : L(k_{−γ} ⊗ E^{∗})] for all E, F ∈ Λ.

Proof. We will here only give a proof under the additional assumption that the simple object L = L(k_{−γ} ⊗ E^{∗}) admits an indecomposable projective cover P = P (k_{−γ}⊗E^{∗}) in O. (The general case is more or less the same, details can be found in [Soe97a].) Under our assumption one proves as usual that P admits a finite Δ-flag ending with Δ(k_{−γ} ⊗ E^{∗}), and a suitably general form of the reciprocity principle tells us that the multiplicities in this Δ-flag are given by (P : Δ(F)) = [∇(F) : L] for F ∈ Λ.

Now we take a second look at our functor M →^{∼} M^{opp} to realize that it also gives an equivalence M ∩ O →^{∼} (M ∩ O)^{opp}. This in turn means that our functor transforms P into an indecomposable object T that has a Δ-flag starting with Δ(E) and satisfies Ext_{O}^{1}(Δ(F), T) = 0 ∀F ∈ Λ. In other words, P = P (k_{−γ} ⊗ E^{∗}) gets transformed into our tilting module T = T (E), and we can calculate

\begin{aligned}(T (E) : Δ(F)) & = & (P (k_{−γ} ⊗ E^{∗}) : Δ(k_{−γ} ⊗ F^{∗})) \\ & = & [∇(k_{−γ} ⊗ F^{∗}) : L(k_{−γ} ⊗ E^{∗})]. \end{aligned}

5. The case of quantum groups

If we run the abstract formula 4.4 for the special case of affine Lie algebras, we see that the character formulas for tilting modules in negative level are determined by the character formulas for simple modules in positive level and vice versa. Now results of Kazhdan and Lusztig [KL93, KL94] tell us how to relate representations of quantum groups at roots of one to category O at negative level for affine Lie algebras. Putting all this together, we see that the character formulas of [KT95] for simple highest weight modules in positive level lead to character formulas for tilting modules for quantum groups at roots of one.

This however only works fine in simply laced cases. In general we would need the extension of the results of [KT95] to non-integral highest weights, which is still missing from the literature. The (partially conjectural) outcome is a formula expressing the characters of tilting modules in terms of Kazhdan-Lusztig polynomials, see [Soe97a] for details and [Soe97b] for motivation.

In the special case that our quantum group U is “quantized GL(V)”, we have by [Du95], 3.6 the quantized Schur-Weyl duality

H = End_{U}V^{⊗n}

where H denotes the “quantization” of kS_{n}, i.e. the Hecke algebra of the symmetric group. The same arguments as in Section 1 now tell us, in which way the character formulas for indecomposable tilting modules over U determine the dimensions of all simple modules over H. Recently Ryom-Hansen suceeded in obtaining from there the formulas for these dimensions conjectured in [LLT96] and proved by Ariki [Ari96] and Grojnowski, see [Gec] for an overview.

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

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