# Quiver varieties and finite dimensional representations of quantum affine algebras

## 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 ,Reference29Reference44Reference45.

There is a large amount of literature on finite dimensional representations of see for example ;Reference1Reference10Reference18Reference25Reference28 and the references therein. A basic result relevant to us is due to Chari-Pressley Reference11: irreducible finite dimensional representations are classified by an of polynomials, where -tuple is the rank of This result was announced for Yangian earlier by Drinfel’d .Reference15. 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 hyper-Kähler manifolds, called ALE spaces -dimensionalReference29. 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 .Reference47 and Lusztig Reference33, the author has been studying their properties Reference44Reference45. 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 Reference20. If we replace homology group by equivariant group in the case of -homology we get the affine Hecke algebra , instead of as was shown by Kazhdan-Lusztig Reference26 and Ginzburg Reference13. Thus it became natural to conjecture that an equivariant group of the quiver variety gave us the quantum affine algebra -homology After the author wrote .Reference44, 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 )Reference21Reference58. They used the cotangent bundle of the partial flag variety, which is an example of a quiver variety of type -step Thus their result can be considered as a partial solution to the conjecture. .

In Reference23 Grojnowski constructed the lower-half part of on equivariant of a certain lagrangian subvariety of the cotangent bundle of a variety -homology 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 .Reference52: a homomorphism from the finite Hecke algebra to the equivariant of the Steinberg variety is defined by assigning to perverse sheaves (or more precisely Hodge modules) on -homology their characteristic cycles. In the same way, he considered characteristic cycles of perverse sheaves on Thus he obtained a homomorphism from . to 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. -homology

Later Grojnowski wrote an ‘advertisement’ of his book on the full conjecture Reference24. 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 .Reference26 gave a classification of simple modules of using the above mentioned , construction. Our analogue is the Drinfel’d-Chari-Pressley classification. Also Ginzburg gave a character formula, called a -theoretic analogue of the Kazhdan-Lusztig multiplicity formula -adicReference13. (See the introduction in Reference13 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 ,Reference15. 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 Reference12 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 where -action, .

- (4)
There is a projective morphism -equivariant .

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

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 structure on -algebra determined by By the convolution product, . has a structure. Thus it has a -module structure by the above homomorphism. By the localization theorem of equivariant -module due to Thomason -homologyReference55, 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 Reference33, 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 -charactersReference18.

- (2)
Decompose restrictions of finite dimensional to -modules (see -modulesReference28).

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

Frenkel-Reshetikhin’s are generating functions of dimensions of -characters*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 .Reference18 for of type (Proposition 13.5.2). These Euler numbers should be computable.

Let be the restriction of to a In § -module.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 Reference44, 8.4. The Poincaré polynomials of were calculated by Lusztig Reference30 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 Reference9 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 Reference22Reference48Reference49Reference56 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 Reference57. 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 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 -dimensionalReference46). We will return to this in the future.

If we replace equivariant by equivariant homology, we should get the Yangian -homology instead of This conjecture is motivated again by the analogy of quiver varieties with . The equivariant homology of . gives the *graded* Hecke algebra Reference32, 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 Reference17 proved the conjecture in Reference18 (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 (weight lattice), -module

- (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 generated by -algebra , ( ), ( with relations )

where , .

Let (resp. be the ) of -subalgebra generated by the elements (resp. Let ). be the generated by elements -subalgebra ( Then we have the triangle decomposition ).Reference36, 3.2.5:

Let and Let . be the of -subalgebra generated by elements , , for , , It is known that . is an *integral form* of i.e., the natural map , is an isomorphism. (See Reference10, 9.3.1.) For let us define , as via the algebra homomorphism that takes to It will be called the .*specialized* quantized enveloping algebra. We say a -module (defined over is a )*highest weight module* with *highest weight* if there exists a vector such that

Then there exists a direct sum decomposition (weight space decomposition) where for any By using the triangular decomposition }.Equation1.1.6, one can show that the simple highest weight is determined uniquely by -module .

We say a -module (defined over is )*integrable* if has a weight space decomposition with and for any , there exists , such that for all and .

The (unique) simple highest weight with highest weight -module is integrable if and only if is a dominant integral weight i.e., , for any (Reference36, 3.5.6, 3.5.8). In this case, the integrable highest weight with highest weight -module is denoted by .

For a -module (defined over we define highest weight modules, integrable modules, etc. in a similar way. ),

Suppose is dominant. Let where , is the highest weight vector. It is known that the natural map is an isomorphism and is the simple integrable highest weight module of the corresponding Kac-Moody algebra with highest weight where , is the homomorphism that sends to (Reference36, Chapter 14 and 33.1.3). Unless is a root of unity, the simple integrable highest weight is the specialization of -module (Reference10, 10.1.14, 10.1.15).

### 1.2. Quantum affine algebra

The *quantum affinization* of (or simply *quantum affine algebra*) is an associative algebra over generated by ( , ), ( ), , and , ( , with the following defining relations: )

where , , , and , is the symmetric group of letters. Here , , , are generating functions defined by

We will also need the following generating function later:

We have

#### Remark 1.2.12

When is finite dimensional, then Then the relation .Equation1.2.10 reduces to the one in literature. Our generalization seems natural since we will check it later, at least for symmetric .

Let (resp. be the ) of -subalgebra generated by the elements (resp. Let ). be the generated by the elements -subalgebra , .

The *quantum loop algebra* is the subalgebra of generated by ( , ), ( and ), ( , i.e., generators other than ), , We will be concerned only with the quantum loop algebra, and not with the quantum affine algebra in the sequel. .

There is a homomorphism defined by

Let and Let . be the generated by -subalgebra , , and the coefficients of for , , , (It should be true that . is free over and that the natural map is an isomorphism. But the author does not know how to prove this.) This subalgebra was introduced by Chari-Pressley Reference12. Let (resp. be the ) generated by -subalgebra (resp. for ) , , We have . Let . be the generated by -subalgebra the coefficients of , and

for all , , , One can easily show that . (see, e.g., Reference36, 3.1.9).

For let , be the *specialized quantum loop algebra* defined by via the algebra homomorphism that takes to We assume . is *not* a root of unity in this paper. Let and be the specializations of and respectively. We have a weak form of the triangular decomposition

which follows from the definition (cf. Reference12, 6.1).

We say a -module is an *l-highest weight module* (‘*l*’ stands for the loop) with *l-highest weight* (where , if there exists a vector ) such that

By using Equation1.2.13 and a standard argument, one can show that there is a simple *l*-highest weight module of with *l*-highest weight vector satisfying the above for any with , Moreover, such . is unique up to isomorphism. For abuse of notation, we denote the pair simply by the symbol .

A -module is said to be *l-integrable* if

- (a)
has a weight space decomposition as a such that -module ,

- (b)
for any there exists , such that for all , and .

For example, if is finite dimensional, and is a finite dimensional module, then satisfies the above conditions after twisting with a certain automorphism of (Reference10, 12.2.3).

#### Proposition 1.2.16

Assume that is symmetric. The simple *l*-highest weight -module with *l*-highest weight is *l*-integrable if and only if is dominant and there exist polynomials for with such that

where and , denotes the expansion at and respectively.

This result was announced by Drinfel’d for the Yangian Reference15. The proof of the ‘only if’ part when is finite dimensional was given by Chari-Pressley Reference10, 12.2.6. Since the proof is based on a reduction to the case it can be applied to a general Kac-Moody algebra , (not necessarily symmetric). The ‘if’ part was proved by them later in Reference11 when is finite dimensional, again not necessarily symmetric. As an application of the main result of this paper, we will prove the converse for a symmetric Kac-Moody algebra in §13. Our proof is independent of Chari-Pressley’s proof.

#### Remark 1.2.18

The polynomials are called *Drinfel’d polynomials*.

When the Drinfel’d polynomials are given by

for some , the corresponding simple ,*l*-highest weight module is called an *l-fundamental representation*. When is finite dimensional, is a Hopf algebra since Drinfel’d Reference15 announced and Beck Reference5 proved that can be identified with (a quotient of) the specialized quantized enveloping algebra associated with Cartan data of affine type. Thus a tensor product of is again a -modules We have the following: -module.

#### Proposition 1.2.19 (Reference10, 12.2.6,12.2.8).

Suppose is finite dimensional.

(1) If and are simple *l*-highest weight with Drinfel’d polynomials -modules , such that is simple, then its Drinfel’d polynomial is given by

(2) Every simple *l*-highest weight is a subquotient of a tensor product of -module*l*-fundamental representations.

Unfortunately the coproduct is not defined for general as far as the author knows. Thus the above results do not make sense for general .

### 1.3. An *l*-weight space decomposition

Let be an *l*-integrable with the weight space decomposition -module Since the commutative subalgebra . preserves each we can further decompose , into a sum of generalized simultaneous eigenspaces for :

where is a pair as before and

If we call , an *l-weight space*, and the corresponding an *l-weight*. This is a refinement of the weight space decomposition. A further study of the *l*-weight space decomposition will be given in §13.5.

Motivated by Proposition 1.2.16, we introduce the following notion:

#### Definition 1.3.2

An *l*-weight is said to be * l-dominant* if is dominant and there exists a polynomial for with such that Equation1.2.17 holds.

Thus Proposition 1.2.16 means that an *l*-highest weight module is *l*-integrable if and only if the *l*-highest weight is *l*-dominant.

## 2. Quiver variety

### 2.1. Notation

Suppose that a finite graph is given and assume that there are no edge loops, i.e., no edges joining a vertex with itself. Let be the set of vertices and the set of edges. Let be the adjacency matrix of the graph, namely

We associate with the graph a symmetric generalized Cartan matrix where , is the identity matrix. This gives a bijection between the finite graphs without edge loops and symmetric Cartan matrices. We have the corresponding symmetric Kac-Moody algebra the quantized enveloping algebra , the quantum affine algebra , and the quantum loop algebra Let . be the set of pairs consisting of an edge together with its orientation. For we denote by , (resp. the incoming (resp. outgoing) vertex of ) For . we denote by the same edge as with the reverse orientation. Choose and fix an orientation of the graph, i.e., a subset such that , The pair . is called a *quiver*. Let us define matrices and by

So we have , .

Let be a collection of finite-dimensional vector spaces over for each vertex The dimension of . is a vector

If and are such collections, we define vector spaces by

For and let us define a multiplication of , and by

Multiplications , of , , are defined in an obvious manner. If its trace , is understood as .

For two collections , of vector spaces with , we consider the vector space given by ,

where we use the notation unless we want to specify dimensions of , The above three components for an element of . will be denoted by , , respectively. An element of will be called an *ADHM datum*.

Usually a point in is called a *representation of the quiver* in the literature. Thus is the product of the space of representations of and that of On the other hand, the factor . or has never appeared in the literature.

#### Convention 2.1.4

When we relate the quiver varieties to the quantum affine algebra, the dimension vectors will be mapped into the weight lattice in the following way:

where (resp. is the ) component of th (resp. Since ). and are both linearly independent, these maps are injective. We consider and as elements of the weight lattice in this way hereafter.

For a collection of subspaces of and we say , is * -invariant* if .

Fix a function such that for all In .Reference44Reference45, it was assumed that takes its value but this assumption is not necessary as remarked by Lusztig ,Reference38. For let us denote by , data given by for .

Let us define a symplectic form on by