Quiver varieties and finite dimensional representations of quantum affine algebras

By Hiraku Nakajima

Abstract

We study finite dimensional representations of the quantum affine algebra using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.

Introduction

Let be a simple finite dimensional Lie algebra of type , let be the corresponding (untwisted) affine Lie algebra, and let be its quantum enveloping algebra of Drinfel’d-Jimbo, or the quantum affine algebra for short. In this paper we study finite dimensional representations of , using geometry of quiver varieties which were introduced in Reference 29Reference 44Reference 45.

There is a large amount of literature on finite dimensional representations of ; see for example Reference 1Reference 10Reference 18Reference 25Reference 28 and the references therein. A basic result relevant to us is due to Chari-Pressley Reference 11: irreducible finite dimensional representations are classified by an -tuple of polynomials, where is the rank of . This result was announced for Yangian earlier by Drinfel’d Reference 15. Hence the polynomials are called Drinfel’d polynomials. However, not much is known about the properties of irreducible finite dimensional representations, say their dimensions, tensor product decomposition, etc.

Quiver varieties are generalizations of moduli spaces of instantons (anti-self-dual connections) on certain classes of real -dimensional hyper-Kähler manifolds, called ALE spaces Reference 29. They can be defined for any finite graph, but we are concerned for the moment with the Dynkin graph of type corresponding to . Motivated by results of Ringel Reference 47 and Lusztig Reference 33, the author has been studying their properties Reference 44Reference 45. In particular, it was shown that there is a homomorphism

where is the universal enveloping algebra of , is a certain lagrangian subvariety of the product of quiver varieties (the quiver variety depends on a choice of a dominant weight ), and denotes the top degree homology group with complex coefficients. The multiplication on the right hand side is defined by the convolution product.

During the study, it became clear that the quiver varieties are analogous to the cotangent bundle of the flag variety . The lagrangian subvariety is an analogue of the Steinberg variety , where is the nilpotent cone and is the Springer resolution. The above mentioned result is an analogue of Ginzburg’s lagrangian construction of the Weyl group Reference 20. If we replace homology group by equivariant -homology group in the case of , we get the affine Hecke algebra instead of as was shown by Kazhdan-Lusztig Reference 26 and Ginzburg Reference 13. Thus it became natural to conjecture that an equivariant -homology group of the quiver variety gave us the quantum affine algebra . After the author wrote Reference 44, many people suggested this conjecture to him, for example Kashiwara, Ginzburg, Lusztig and Vasserot.

A geometric approach to finite dimensional representations of (when ) was given by Ginzburg-Vasserot Reference 21Reference 58. They used the cotangent bundle of the -step partial flag variety, which is an example of a quiver variety of type . Thus their result can be considered as a partial solution to the conjecture.

In Reference 23 Grojnowski constructed the lower-half part of on equivariant -homology of a certain lagrangian subvariety of the cotangent bundle of a variety . This was used earlier by Lusztig for the construction of canonical bases on the lower-half part of the quantized enveloping algebra . Grojnowski’s construction was motivated in part by Tanisaki’s result Reference 52: a homomorphism from the finite Hecke algebra to the equivariant -homology of the Steinberg variety is defined by assigning to perverse sheaves (or more precisely Hodge modules) on their characteristic cycles. In the same way, he considered characteristic cycles of perverse sheaves on . Thus he obtained a homomorphism from to -homology of the lagrangian subvariety. This lagrangian subvariety contains a lagrangian subvariety of the quiver variety as an open subvariety. Thus his construction was a solution to ‘half’ of the conjecture.

Later Grojnowski wrote an ‘advertisement’ of his book on the full conjecture Reference 24. Unfortunately, details were not explained, and his book is not published yet.

The purpose of this paper is to solve the conjecture affirmatively, and to derive results whose analogues are known for . Recall that Kazhdan-Lusztig Reference 26 gave a classification of simple modules of , using the above mentioned -theoretic construction. Our analogue is the Drinfel’d-Chari-Pressley classification. Also Ginzburg gave a character formula, called a -adic analogue of the Kazhdan-Lusztig multiplicity formula Reference 13. (See the introduction in Reference 13 for a more detailed account and historical comments.) We prove a similar formula for in this paper.

Let us describe the contents of this paper in more detail. In §1 we recall a new realization of , called Drinfel’d realization Reference 15. It is more suitable than the original one for our purpose, or rather, we can consider it as a definition of . We also introduce the quantum loop algebra , which is a subquotient of , i.e., the quantum affine algebra without central extension and the degree operator. Since the central extension acts trivially on finite dimensional representations, we study rather than . Introducing a certain -subalgebra of , we define a specialization of at . This was originally introduced by Chari-Pressley Reference 12 for the study of finite dimensional representations of when is a root of unity. Then we recall basic results on finite dimensional representations of . We introduce several concepts, such as l-weights, l-dominant, l-highest weight modules, l-fundamental representation, etc. These are analogues of the same concepts without l for -modules.l’ stands for the loop. In the literature, some of these concepts were used without ‘l’.

In §2 we introduce two types of quiver varieties, , (both depend on a choice of a dominant weight ). They are analogues of and the nilpotent cone respectively, and have the following properties:

(1)

is a nonsingular quasi-projective variety, having many components of various dimensions.

(2)

is an affine algebraic variety, not necessarily irreducible.

(3)

Both and have a -action, where .

(4)

There is a -equivariant projective morphism .

In §3–§8 we prepare some results on quiver varieties and -theory which we use in later sections.

In §9–§11 we consider an analogue of the Steinberg variety and its equivariant -homology . We construct an algebra homomorphism

We first define images of generators in §9, and check the defining relations in §10 and §11. Unlike the case of the affine Hecke algebra, where is isomorphic to ( the Steinberg variety), this homomorphism is not an isomorphism, neither injective nor surjective.

In §12 we show that the above homomorphism induces a homomorphism

(It is natural to expect that is an integral form of and that is torsion-free, but we do not have the proofs.)

In §13 we introduce a standard module . It depends on the choice of a point and a semisimple element such that is fixed by . The parameter corresponds to the specialization , while corresponds to Drinfel’d polynomials. In this paper, we assume is not a root of unity, although most of our results hold even in that case (see Remark 14.3.9). Let be the Zariski closure of . We define as the specialized equivariant -homology , where is a fiber of at , and is an -algebra structure on determined by . By the convolution product, has a -module structure. Thus it has a -module structure by the above homomorphism. By the localization theorem of equivariant -homology due to Thomason Reference 55, is isomorphic to the complexified (non-equivariant) -homology of the fixed point set . Moreover, it is isomorphic to via the Chern character homomorphism thanks to a result in §7. We also show that is a finite dimensional l-highest weight module. As a usual argument for Verma modules, has the unique (nonzero) simple quotient. The author conjectures that is a tensor product of l-fundamental representations in some order. This is proved when the parameter is generic in §14.1.

In §14 we show that the standard modules and are isomorphic if and only if and are contained in the same stratum. Here the fixed point set has a stratification defined in §4. Furthermore, we show that the index set of the stratum coincides with the set of l-dominant l-weights of , the standard module corresponding to the central fiber . Let us denote by the index corresponding to . Thus we may denote and its unique simple quotient by and respectively if is contained in the stratum corresponding to an l-dominant l-weight . We prove the multiplicity formula

where is a point in , is the inclusion, and is the intersection cohomology complex attached to and the constant local system .

Our result is simpler than the case of the affine Hecke algebra: nonconstant local systems never appear. This phenomenon corresponds to an algebraic result that all modules are l-highest weight. It compensates for the difference of and during the proof of the multiplicity formula.

If is of type , then coincides with a product of varieties studied by Lusztig Reference 33, where the underlying graph is of type . In particular, the Poincaré polynomial of is a Kazhdan-Lusztig polynomial for a Weyl group of type . We should have a combinatorial algorithm to compute Poincaré polynomials of for general .

Once we know , information about can be deduced from information about , which is easier to study. For example, consider the following problems:

(1)

Compute Frenkel-Reshetikhin’s -characters Reference 18.

(2)

Decompose restrictions of finite dimensional -modules to -modules (see Reference 28).

These problems for are easier than those for , and we have the following answers.

Frenkel-Reshetikhin’s -characters are generating functions of dimensions of l-weight spaces (see §13.5). In §13.5 we show that these dimensions are Euler numbers of connected components of for standard modules . As an application, we prove a conjecture in Reference 18 for of type (Proposition 13.5.2). These Euler numbers should be computable.

Let be the restriction of to a -module. In §15 we show the multiplicity formula

where is a weight such that is dominant, is the corresponding irreducible finite dimensional module (these are concepts for usual without ‘l’), is a point in , is the inclusion, is a stratum of , and is the intersection cohomology complex attached to and the constant local system .

If is of type , then the stratum coincides with a nilpotent orbit cut out by Slodowy’s transversal slice Reference 44, 8.4. The Poincaré polynomials of were calculated by Lusztig Reference 30 and coincide with Kostka polynomials. This result is compatible with the conjecture that is a tensor product of l-fundamental representations, for the restriction of an l-fundamental representation is simple for type , and Kostka polynomials give tensor product decompositions. We should have a combinatorial algorithm to compute Poincaré polynomials of for general .

We give two examples where can be described explicitly.

Consider the case that is a fundamental weight of type , or more generally a fundamental weight such that the label of the corresponding vertex of the Dynkin diagram is . Then it is easy to see that the corresponding quiver variety consists of a single point . Thus remains irreducible in this case.

If is the highest weight of the adjoint representation, the corresponding is a simple singularity , where is a finite subgroup of of the type corresponding to . Then has two strata and . The intersection cohomology complexes are constant sheaves. Hence we have

These two results were shown by Chari-Pressley Reference 9 by a totally different method.

As we mentioned, the quantum affine algebra has another realization, called the Drinfel’d new realization. This Drinfel’d construction can be applied to any symmetrizable Kac-Moody algebra , not necessarily a finite dimensional one. This generalization also fits our result, since quiver varieties can be defined for arbitrary finite graphs. If we replace finite dimensional representations by l-integrable representations, parts of our result can be generalized to a Kac-Moody algebra , at least when it is symmetric. For example, we generalize the Drinfel’d-Chari-Pressley parametrization. A generalization of the multiplicity formula requires further study.

If is an affine Lie algebra, then is the quantum affinization of the affine Lie algebra. It is called a double loop algebra, or toroidal algebra, and has been studied by various people; see for example Reference 22Reference 48Reference 49Reference 56 and the references therein. A first step to a geometric approach to the toroidal algebra using quiver varieties for the affine Dynkin graph of type was given by M. Varagnolo and E. Vasserot Reference 57. In fact, quiver varieties for affine Dynkin graphs are moduli spaces of instantons (or torsion free sheave) on ALE spaces. Thus these cases are relevant to the original motivation, i.e., a study of the relation between -dimensional gauge theory and representation theory. In some cases, these quiver varieties coincide with Hilbert schemes of points on ALE spaces, for which many results have been obtained (see Reference 46). We will return to this in the future.

If we replace equivariant -homology by equivariant homology, we should get the Yangian instead of . This conjecture is motivated again by the analogy of quiver varieties with . The equivariant homology of gives the graded Hecke algebra Reference 32, which is an analogue of for . As an application, the affirmative solution of the conjecture implies that the representation theory of and that of the Yangian are the same. This has been believed by many people, but there is no written proof.

While the author was preparing this paper, he was informed that Frenkel-Mukhin Reference 17 proved the conjecture in Reference 18 (Proposition 13.5.2) for general .

Acknowledgement.

Part of this work was done while the author enjoyed the hospitality of the Institute for Advanced Study. The author is grateful to G. Lusztig for his interest and encouragement.

1. Quantum affine algebra

In this section, we give a quick review for the definitions of the quantized universal enveloping algebra of the Kac-Moody algebra associated with a symmetrizable generalized Cartan matrix, its affinization , and the associated loop algebra . Although the algebras defined via quiver varieties are automatically symmetric, we treat the nonsymmetric case also for completeness.

1.1. Quantized universal enveloping algebra

Let be an indeterminate. For nonnegative integers , define

Suppose that the following data are given:

(1)

: free -module (weight lattice),

(2)

with a natural pairing ,

(3)

an index set of simple roots

(4)

() (simple root),

(5)

() (simple coroot),

(6)

a symmetric bilinear form on .

These are required to satisfy the following:

(a)

for and ,

(b)

is a symmetrizable generalized Cartan matrix, i.e., , and and for ,

(c)

,

(d)

are linearly independent,

(e)

there exists () such that (fundamental weight).

The quantized universal enveloping algebra of the Kac-Moody algebra is the -algebra generated by , (), () with relations