American Mathematical Society

Quiver varieties and finite dimensional representations of quantum affine algebras

By Hiraku Nakajima

Abstract

We study finite dimensional representations of the quantum affine algebra bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis 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 German g be a simple finite dimensional Lie algebra of type upper A upper D upper E , let ModifyingAbove German g With caret be the corresponding (untwisted) affine Lie algebra, and let bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis 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 bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis , using geometry of quiver varieties which were introduced in Reference29Reference44Reference45.

There is a large amount of literature on finite dimensional representations of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis ; 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 n -tuple of polynomials, where n is the rank of German g . 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 4 -dimensional hyper-Kähler manifolds, called ALE spaces Reference29. They can be defined for any finite graph, but we are concerned for the moment with the Dynkin graph of type upper A upper D upper E corresponding to German g . 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

bold upper U left-parenthesis German g right-parenthesis right-arrow upper H Subscript t o p Baseline left-parenthesis upper Z left-parenthesis bold w right-parenthesis comma double-struck upper C right-parenthesis comma

where bold upper U left-parenthesis German g right-parenthesis is the universal enveloping algebra of German g , upper Z left-parenthesis bold w right-parenthesis is a certain lagrangian subvariety of the product of quiver varieties (the quiver variety depends on a choice of a dominant weight bold w ), and upper H Subscript t o p Baseline left-parenthesis comma double-struck upper C right-parenthesis 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 upper T Superscript asterisk Baseline script upper B of the flag variety script upper B . The lagrangian subvariety upper Z left-parenthesis bold w right-parenthesis is an analogue of the Steinberg variety upper Z equals upper T Superscript asterisk Baseline script upper B times Subscript script upper N Baseline upper T Superscript asterisk Baseline script upper B , where script upper N is the nilpotent cone and upper T Superscript asterisk Baseline script upper B right-arrow script upper N is the Springer resolution. The above mentioned result is an analogue of Ginzburg’s lagrangian construction of the Weyl group upper W Reference20. If we replace homology group by equivariant upper K -homology group in the case of upper T Superscript asterisk Baseline script upper B , we get the affine Hecke algebra upper H Subscript q instead of upper W as was shown by Kazhdan-Lusztig Reference26 and Ginzburg Reference13. Thus it became natural to conjecture that an equivariant upper K -homology group of the quiver variety gave us the quantum affine algebra bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis . 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 bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis (when German g equals German s German l Subscript n ) was given by Ginzburg-Vasserot Reference21Reference58. They used the cotangent bundle of the n -step partial flag variety, which is an example of a quiver variety of type upper A . Thus their result can be considered as a partial solution to the conjecture.

In Reference23 Grojnowski constructed the lower-half part bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis Superscript minus of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis on equivariant upper K -homology of a certain lagrangian subvariety of the cotangent bundle of a variety bold upper E Subscript bold d . This bold upper E Subscript bold d was used earlier by Lusztig for the construction of canonical bases on the lower-half part bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript minus of the quantized enveloping algebra bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis . Grojnowski’s construction was motivated in part by Tanisaki’s result Reference52: a homomorphism from the finite Hecke algebra to the equivariant upper K -homology of the Steinberg variety is defined by assigning to perverse sheaves (or more precisely Hodge modules) on script upper B their characteristic cycles. In the same way, he considered characteristic cycles of perverse sheaves on bold upper E Subscript bold d . Thus he obtained a homomorphism from bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript minus to upper K -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 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 upper H Subscript q . Recall that Kazhdan-Lusztig Reference26 gave a classification of simple modules of upper H Subscript q , using the above mentioned upper K -theoretic construction. Our analogue is the Drinfel’d-Chari-Pressley classification. Also Ginzburg gave a character formula, called a p -adic analogue of the Kazhdan-Lusztig multiplicity formula Reference13. (See the introduction in Reference13 for a more detailed account and historical comments.) We prove a similar formula for bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis in this paper.

Let us describe the contents of this paper in more detail. In §1 we recall a new realization of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis , 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 bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis . We also introduce the quantum loop algebra bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis , which is a subquotient of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis , 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 bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis rather than bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis . Introducing a certain double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket -subalgebra bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis of bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis , we define a specialization bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis of bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis at q equals epsilon . This bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis was originally introduced by Chari-Pressley Reference12 for the study of finite dimensional representations of bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis when epsilon is a root of unity. Then we recall basic results on finite dimensional representations of bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis . 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 bold upper U Subscript epsilon Baseline left-parenthesis German g right-parenthesis -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, German upper M left-parenthesis bold w right-parenthesis , German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis (both depend on a choice of a dominant weight bold w equals sigma-summation w Subscript k Baseline normal upper Lamda Subscript k ). They are analogues of upper T Superscript asterisk Baseline script upper B and the nilpotent cone script upper N respectively, and have the following properties:

(1)

German upper M left-parenthesis bold w right-parenthesis is a nonsingular quasi-projective variety, having many components of various dimensions.

(2)

German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis is an affine algebraic variety, not necessarily irreducible.

(3)

Both German upper M left-parenthesis bold w right-parenthesis and German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis have a upper G Subscript bold w Baseline times double-struck upper C Superscript asterisk -action, where upper G Subscript bold w Baseline equals product upper G upper L Subscript w Sub Subscript k Baseline left-parenthesis double-struck upper C right-parenthesis .

(4)

There is a upper G Subscript bold w Baseline times double-struck upper C Superscript asterisk -equivariant projective morphism pi colon German upper M left-parenthesis bold w right-parenthesis right-arrow German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis .

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

In §9–§11 we consider an analogue of the Steinberg variety upper Z left-parenthesis bold w right-parenthesis equals German upper M left-parenthesis bold w right-parenthesis times German upper M left-parenthesis bold w right-parenthesis and its equivariant upper K -homology upper K Superscript upper G Super Subscript bold w Superscript times double-struck upper C Super Superscript asterisk Baseline left-parenthesis upper Z left-parenthesis bold w right-parenthesis right-parenthesis . We construct an algebra homomorphism

bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis right-arrow upper K Superscript upper G Super Subscript bold w Superscript times double-struck upper C Super Superscript asterisk Superscript Baseline left-parenthesis upper Z left-parenthesis bold w right-parenthesis right-parenthesis circled-times Subscript double-struck upper Z left-bracket q comma q Sub Superscript negative 1 Subscript right-bracket Baseline double-struck upper Q left-parenthesis q right-parenthesis period

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 upper H Subscript q is isomorphic to upper K Superscript upper G times double-struck upper C Super Superscript asterisk Baseline left-parenthesis upper Z right-parenthesis ( upper Z equals the Steinberg variety), this homomorphism is not an isomorphism, neither injective nor surjective.

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

bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis right-arrow upper K Superscript upper G Super Subscript bold w Superscript times double-struck upper C Super Superscript asterisk Superscript Baseline left-parenthesis upper Z left-parenthesis bold w right-parenthesis right-parenthesis slash t o r s i o n period

(It is natural to expect that bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis is an integral form of bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis and that upper K Superscript upper G Super Subscript bold w Superscript times double-struck upper C Super Superscript asterisk Baseline left-parenthesis upper Z left-parenthesis bold w right-parenthesis right-parenthesis is torsion-free, but we do not have the proofs.)

In §13 we introduce a standard module upper M Subscript x comma a . It depends on the choice of a point x element-of German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis and a semisimple element a equals left-parenthesis s comma epsilon right-parenthesis element-of upper G Subscript bold w Baseline times double-struck upper C Superscript asterisk such that x is fixed by a . The parameter epsilon corresponds to the specialization q equals epsilon , while s corresponds to Drinfel’d polynomials. In this paper, we assume epsilon is not a root of unity, although most of our results hold even in that case (see Remark 14.3.9). Let upper A be the Zariski closure of a Superscript double-struck upper Z . We define upper M Subscript x comma a as the specialized equivariant upper K -homology upper K Superscript upper A Baseline left-parenthesis German upper M left-parenthesis bold w right-parenthesis Subscript x Baseline right-parenthesis circled-times Subscript upper R left-parenthesis upper A right-parenthesis Baseline double-struck upper C Subscript a , where German upper M left-parenthesis bold w right-parenthesis Subscript x is a fiber of German upper M left-parenthesis bold w right-parenthesis right-arrow German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis at x , and double-struck upper C Subscript a is an upper R left-parenthesis upper A right-parenthesis -algebra structure on double-struck upper C determined by a . By the convolution product, upper M Subscript x comma a has a upper K Superscript upper A Baseline left-parenthesis upper Z left-parenthesis bold w right-parenthesis right-parenthesis circled-times Subscript upper R left-parenthesis upper A right-parenthesis Baseline double-struck upper C Subscript a -module structure. Thus it has a bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -module structure by the above homomorphism. By the localization theorem of equivariant upper K -homology due to Thomason Reference55, upper M Subscript x comma a is isomorphic to the complexified (non-equivariant) upper K -homology upper K left-parenthesis German upper M left-parenthesis bold w right-parenthesis Subscript x Superscript upper A Baseline right-parenthesis circled-times double-struck upper C of the fixed point set German upper M left-parenthesis bold w right-parenthesis Subscript x Superscript upper A . Moreover, it is isomorphic to upper H Subscript asterisk Baseline left-parenthesis German upper M left-parenthesis bold w right-parenthesis Subscript x Superscript upper A Baseline comma double-struck upper C right-parenthesis via the Chern character homomorphism thanks to a result in §7. We also show that upper M Subscript x comma a is a finite dimensional l-highest weight module. As a usual argument for Verma modules, upper M Subscript x comma a has the unique (nonzero) simple quotient. The author conjectures that upper M Subscript x comma a 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 upper M Subscript x comma a and upper M Subscript y comma a are isomorphic if and only if x and y are contained in the same stratum. Here the fixed point set German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis Superscript upper A has a stratification German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis Superscript upper A Baseline equals square-union Underscript rho Endscripts German upper M 0 Superscript r e g Baseline left-parenthesis rho right-parenthesis defined in §4. Furthermore, we show that the index set StartSet rho EndSet of the stratum coincides with the set script upper P equals StartSet upper P EndSet of l-dominant l-weights of upper M Subscript 0 comma a , the standard module corresponding to the central fiber pi Superscript negative 1 Baseline left-parenthesis 0 right-parenthesis . Let us denote by rho Subscript upper P the index corresponding to upper P . Thus we may denote upper M Subscript x comma a and its unique simple quotient by upper M left-parenthesis upper P right-parenthesis and upper L left-parenthesis upper P right-parenthesis respectively if x is contained in the stratum German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper P Baseline right-parenthesis corresponding to an l-dominant l-weight upper P . We prove the multiplicity formula

left-bracket upper M left-parenthesis upper P right-parenthesis colon upper L left-parenthesis upper Q right-parenthesis right-bracket equals dimension upper H Superscript asterisk Baseline left-parenthesis i Subscript x Superscript factorial Baseline upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper Q Baseline right-parenthesis right-parenthesis right-parenthesis comma

where x is a point in German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper P Baseline right-parenthesis , i Subscript x Baseline colon StartSet x EndSet right-arrow German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis Superscript upper A is the inclusion, and upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper Q Baseline right-parenthesis right-parenthesis is the intersection cohomology complex attached to German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper Q Baseline right-parenthesis and the constant local system double-struck upper C Subscript German upper M 0 Sub Superscript r e g Subscript left-parenthesis rho Sub Subscript upper Q Subscript right-parenthesis .

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 bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis and upper K Superscript upper G Super Subscript bold w Superscript times double-struck upper C Super Superscript asterisk Baseline left-parenthesis upper Z left-parenthesis bold w right-parenthesis right-parenthesis during the proof of the multiplicity formula.

If German g is of type upper A , then German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis Superscript upper A coincides with a product of varieties bold upper E Subscript bold d studied by Lusztig Reference33, where the underlying graph is of type upper A . In particular, the Poincaré polynomial of upper H Superscript asterisk Baseline left-parenthesis i Subscript x Superscript factorial Baseline upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper Q Baseline right-parenthesis right-parenthesis right-parenthesis is a Kazhdan-Lusztig polynomial for a Weyl group of type upper A . We should have a combinatorial algorithm to compute Poincaré polynomials of upper H Superscript asterisk Baseline left-parenthesis i Subscript x Superscript factorial Baseline upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper Q Baseline right-parenthesis right-parenthesis right-parenthesis for general German g .

Once we know dimension upper H Superscript asterisk Baseline left-parenthesis i Subscript x Superscript factorial Baseline upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper Q Baseline right-parenthesis right-parenthesis right-parenthesis , information about upper L left-parenthesis upper P right-parenthesis can be deduced from information about upper M left-parenthesis upper P right-parenthesis , which is easier to study. For example, consider the following problems:

(1)

Compute Frenkel-Reshetikhin’s q -characters Reference18.

(2)

Decompose restrictions of finite dimensional bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -modules to bold upper U Subscript epsilon Baseline left-parenthesis German g right-parenthesis -modules (see Reference28).

These problems for upper M left-parenthesis upper P right-parenthesis are easier than those for upper L left-parenthesis upper P right-parenthesis , and we have the following answers.

Frenkel-Reshetikhin’s q -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 German upper M left-parenthesis bold w right-parenthesis Superscript upper A for standard modules upper M Subscript 0 comma a . As an application, we prove a conjecture in Reference18 for German g of type upper A upper D upper E (Proposition 13.5.2). These Euler numbers should be computable.

Let upper R e s upper M left-parenthesis upper P right-parenthesis be the restriction of upper M left-parenthesis upper P right-parenthesis to a bold upper U Subscript epsilon Baseline left-parenthesis German g right-parenthesis -module. In §15 we show the multiplicity formula

left-bracket upper R e s upper M left-parenthesis upper P right-parenthesis colon upper L left-parenthesis bold w minus bold v right-parenthesis right-bracket equals dimension upper H Superscript asterisk Baseline left-parenthesis i Subscript x Superscript factorial Baseline upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis right-parenthesis right-parenthesis comma

where bold v is a weight such that bold w minus bold v is dominant, upper L left-parenthesis bold w minus bold v right-parenthesis is the corresponding irreducible finite dimensional module (these are concepts for usual German g without ‘l’), x is a point in German upper M 0 Superscript r e g Baseline left-parenthesis rho Subscript upper P Baseline right-parenthesis , i Subscript x Baseline colon StartSet x EndSet right-arrow German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis is the inclusion, German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis is a stratum of German upper M left-parenthesis normal infinity comma bold w right-parenthesis , and upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis right-parenthesis is the intersection cohomology complex attached to German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis and the constant local system double-struck upper C Subscript German upper M 0 Sub Superscript r e g Subscript left-parenthesis bold v comma bold w right-parenthesis .

If German g is of type upper A , then the stratum German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis coincides with a nilpotent orbit cut out by Slodowy’s transversal slice Reference44, 8.4. The Poincaré polynomials of upper H Superscript asterisk Baseline left-parenthesis i Subscript x Superscript factorial Baseline upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis right-parenthesis right-parenthesis were calculated by Lusztig Reference30 and coincide with Kostka polynomials. This result is compatible with the conjecture that upper M left-parenthesis upper P right-parenthesis is a tensor product of l-fundamental representations, for the restriction of an l-fundamental representation is simple for type upper A , and Kostka polynomials give tensor product decompositions. We should have a combinatorial algorithm to compute Poincaré polynomials of upper H Superscript asterisk Baseline left-parenthesis i Subscript x Superscript factorial Baseline upper I upper C left-parenthesis German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis right-parenthesis right-parenthesis for general German g .

We give two examples where German upper M 0 Superscript r e g Baseline left-parenthesis bold v comma bold w right-parenthesis can be described explicitly.

Consider the case that bold w is a fundamental weight of type upper A , or more generally a fundamental weight such that the label of the corresponding vertex of the Dynkin diagram is 1 . Then it is easy to see that the corresponding quiver variety German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis consists of a single point 0 . Thus upper R e s upper M left-parenthesis upper P right-parenthesis remains irreducible in this case.

If bold w is the highest weight of the adjoint representation, the corresponding German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis is a simple singularity double-struck upper C squared slash normal upper Gamma , where normal upper Gamma is a finite subgroup of upper S upper L 2 left-parenthesis double-struck upper C right-parenthesis of the type corresponding to German g . Then German upper M 0 left-parenthesis normal infinity comma bold w right-parenthesis has two strata StartSet 0 EndSet and left-parenthesis double-struck upper C squared minus StartSet 0 EndSet right-parenthesis slash normal upper Gamma . The intersection cohomology complexes are constant sheaves. Hence we have

upper R e s upper M left-parenthesis upper P right-parenthesis equals upper L left-parenthesis bold w right-parenthesis circled-plus upper L left-parenthesis 0 right-parenthesis period

These two results were shown by Chari-Pressley Reference9 by a totally different method.

As we mentioned, the quantum affine algebra bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis has another realization, called the Drinfel’d new realization. This Drinfel’d construction can be applied to any symmetrizable Kac-Moody algebra German g , 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 German g , 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 German g is an affine Lie algebra, then bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis 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 upper A overTilde 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 4 -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 Reference46). We will return to this in the future.

If we replace equivariant upper K -homology by equivariant homology, we should get the Yangian upper Y left-parenthesis German g right-parenthesis instead of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis . This conjecture is motivated again by the analogy of quiver varieties with upper T Superscript asterisk Baseline script upper B . The equivariant homology of upper T Superscript asterisk Baseline script upper B gives the graded Hecke algebra Reference32, which is an analogue of upper Y left-parenthesis German g right-parenthesis for upper H Subscript q . As an application, the affirmative solution of the conjecture implies that the representation theory of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis 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 German g .

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 bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis of the Kac-Moody algebra German g associated with a symmetrizable generalized Cartan matrix, its affinization bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis , and the associated loop algebra bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis . 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 q be an indeterminate. For nonnegative integers n greater-than-or-equal-to r , define

StartLayout 1st Row with Label left-parenthesis 1.1 .1 right-parenthesis EndLabel StartLayout 1st Row 1st Column left-bracket n right-bracket Subscript q Baseline equals Overscript normal d normal e normal f period Endscripts StartFraction q Superscript n Baseline minus q Superscript negative n Baseline Over q minus q Superscript negative 1 Baseline EndFraction comma left-bracket n right-bracket Subscript q Baseline factorial 2nd Column equals Overscript normal d normal e normal f period Endscripts StartLayout Enlarged left-brace 1st Row 1st Column left-bracket n right-bracket Subscript q Baseline left-bracket n minus 1 right-bracket Subscript q Baseline midline-horizontal-ellipsis left-bracket 2 right-bracket Subscript q Baseline left-bracket 1 right-bracket Subscript q Baseline 2nd Column left-parenthesis n greater-than 0 right-parenthesis comma 2nd Row 1st Column 1 2nd Column left-parenthesis n equals 0 right-parenthesis comma EndLayout 2nd Row 1st Column StartBinomialOrMatrix n Choose r EndBinomialOrMatrix Subscript q 2nd Column equals Overscript normal d normal e normal f period Endscripts StartFraction left-bracket n right-bracket Subscript q Baseline factorial Over left-bracket r right-bracket Subscript q Baseline factorial left-bracket n minus r right-bracket Subscript q Baseline factorial EndFraction period EndLayout EndLayout

Suppose that the following data are given:

(1)

upper P : free double-struck upper Z -module (weight lattice),

(2)

upper P Superscript asterisk Baseline equals upper H o m Subscript double-struck upper Z Baseline left-parenthesis upper P comma double-struck upper Z right-parenthesis with a natural pairing mathematical left-angle comma mathematical right-angle colon upper P circled-times upper P Superscript asterisk Baseline right-arrow double-struck upper Z ,

(3)

an index set upper I of simple roots

(4)

alpha Subscript k Baseline element-of upper P ( k element-of upper I ) (simple root),

(5)

h Subscript k Baseline element-of upper P Superscript asterisk ( k element-of upper I ) (simple coroot),

(6)

a symmetric bilinear form left-parenthesis comma right-parenthesis on upper P .

These are required to satisfy the following:

(a)

mathematical left-angle h Subscript k Baseline comma lamda mathematical right-angle equals 2 left-parenthesis alpha Subscript k Baseline comma lamda right-parenthesis slash left-parenthesis alpha Subscript k Baseline comma alpha Subscript k Baseline right-parenthesis for k element-of upper I and lamda element-of upper P ,

(b)

bold upper C equals Overscript normal d normal e normal f period Endscripts left-parenthesis mathematical left-angle h Subscript k Baseline comma alpha Subscript l Baseline mathematical right-angle right-parenthesis Subscript k comma l is a symmetrizable generalized Cartan matrix, i.e., mathematical left-angle h Subscript k Baseline comma alpha Subscript k Baseline mathematical right-angle equals 2 , and mathematical left-angle h Subscript k Baseline comma alpha Subscript l Baseline mathematical right-angle element-of double-struck upper Z Subscript less-than-or-equal-to 0 and mathematical left-angle h Subscript k Baseline comma alpha Subscript l Baseline mathematical right-angle equals 0 long left right double arrow mathematical left-angle h Subscript l Baseline comma alpha Subscript k Baseline mathematical right-angle equals 0 for k not-equals l ,

(c)

left-parenthesis alpha Subscript k Baseline comma alpha Subscript k Baseline right-parenthesis element-of 2 double-struck upper Z Subscript greater-than 0 ,

(d)

StartSet alpha Subscript k Baseline EndSet Subscript k element-of upper I are linearly independent,

(e)

there exists normal upper Lamda Subscript k Baseline element-of upper P ( k element-of upper I ) such that mathematical left-angle h Subscript l Baseline comma normal upper Lamda Subscript k Baseline mathematical right-angle equals delta Subscript k l (fundamental weight).

The quantized universal enveloping algebra bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis of the Kac-Moody algebra is the double-struck upper Q left-parenthesis q right-parenthesis -algebra generated by e Subscript k , f Subscript k ( k element-of upper I ), q Superscript h ( h element-of upper P Superscript asterisk ) with relations

StartLayout 1st Row with Label left-parenthesis 1.1 .2 right-parenthesis EndLabel q Superscript 0 Baseline equals 1 comma q Superscript h Baseline q Superscript h Super Superscript prime Superscript Baseline equals q Superscript h plus h Super Superscript prime Superscript Baseline comma 2nd Row with Label left-parenthesis 1.1 .3 right-parenthesis EndLabel q Superscript h Baseline e Subscript k Baseline q Superscript negative h Baseline equals q Superscript mathematical left-angle h comma alpha Super Subscript k Superscript mathematical right-angle Baseline e Subscript k Baseline comma q Superscript h Baseline f Subscript k Baseline q Superscript negative h Baseline equals q Superscript minus mathematical left-angle h comma alpha Super Subscript k Superscript mathematical right-angle Baseline f Subscript k Baseline comma 3rd Row with Label left-parenthesis 1.1 .4 right-parenthesis EndLabel e Subscript k Baseline f Subscript l Baseline minus f Subscript l Baseline e Subscript k Baseline equals delta Subscript k l Baseline StartFraction q Superscript left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis h Super Subscript k Superscript slash 2 Baseline minus q Superscript minus left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis h Super Subscript k Superscript slash 2 Baseline Over q Subscript k Baseline minus q Subscript k Superscript negative 1 Baseline EndFraction comma 4th Row with Label left-parenthesis 1.1 .5 right-parenthesis EndLabel sigma-summation Underscript p equals 0 Overscript b Endscripts left-parenthesis negative 1 right-parenthesis Superscript p Baseline StartBinomialOrMatrix b Choose p EndBinomialOrMatrix Subscript q Sub Subscript k Baseline e Subscript k Superscript p Baseline e Subscript l Baseline e Subscript k Superscript b minus p Baseline equals sigma-summation Underscript p equals 0 Overscript b Endscripts left-parenthesis negative 1 right-parenthesis Superscript p Baseline StartBinomialOrMatrix b Choose p EndBinomialOrMatrix Subscript q Sub Subscript k Baseline f Subscript k Superscript p Baseline f Subscript l Baseline f Subscript k Superscript b minus p Baseline equals 0 for k not-equals l comma EndLayout

where q Subscript k Baseline equals q Superscript left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis slash 2 , b equals 1 minus mathematical left-angle h Subscript k Baseline comma alpha Subscript l Baseline mathematical right-angle .

Let bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript plus (resp. bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript minus ) be the double-struck upper Q left-parenthesis q right-parenthesis -subalgebra of bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis generated by the elements e Subscript k (resp. f Subscript k ). Let bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript 0 be the double-struck upper Q left-parenthesis q right-parenthesis -subalgebra generated by elements q Superscript h ( h element-of upper P Superscript asterisk ). Then we have the triangle decomposition Reference36, 3.2.5:

StartLayout 1st Row with Label left-parenthesis 1.1 .6 right-parenthesis EndLabel bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis approximately-equals bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript plus Baseline circled-times bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript 0 Baseline circled-times bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript minus Baseline period EndLayout

Let e Subscript k Superscript left-parenthesis n right-parenthesis Baseline equals Overscript normal d normal e normal f period Endscripts e Subscript k Superscript n Baseline slash left-bracket n right-bracket Subscript q Sub Subscript k Subscript Baseline factorial and f Subscript k Superscript left-parenthesis n right-parenthesis Baseline equals Overscript normal d normal e normal f period Endscripts f Subscript k Superscript n Baseline slash left-bracket n right-bracket Subscript q Sub Subscript k Subscript Baseline factorial . Let bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis German g right-parenthesis be the double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket -subalgebra of bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis generated by elements e Subscript k Superscript left-parenthesis n right-parenthesis , f Subscript k Superscript left-parenthesis n right-parenthesis , q Superscript h for k element-of upper I , n element-of double-struck upper Z Subscript greater-than 0 , h element-of upper P Superscript asterisk . It is known that bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis German g right-parenthesis is an integral form of bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis , i.e., the natural map bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis German g right-parenthesis circled-times Subscript double-struck upper Z left-bracket q comma q Sub Superscript negative 1 Subscript right-bracket Baseline double-struck upper Q left-parenthesis q right-parenthesis right-arrow bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis is an isomorphism. (See Reference10, 9.3.1.) For epsilon element-of double-struck upper C Superscript asterisk , let us define bold upper U Subscript epsilon Baseline left-parenthesis German g right-parenthesis as bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis German g right-parenthesis circled-times Subscript double-struck upper Z left-bracket q comma q Sub Superscript negative 1 Subscript right-bracket Baseline double-struck upper C via the algebra homomorphism double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket right-arrow double-struck upper C that takes q to epsilon . It will be called the specialized quantized enveloping algebra. We say a bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis -module upper M (defined over double-struck upper Q left-parenthesis q right-parenthesis ) is a highest weight module with highest weight normal upper Lamda element-of upper P if there exists a vector m 0 element-of upper M such that

StartLayout 1st Row with Label left-parenthesis 1.1 .7 right-parenthesis EndLabel 1st Column Blank 2nd Column e Subscript k Baseline asterisk m 0 equals 0 comma bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis Superscript minus Baseline asterisk m 0 equals upper M comma 2nd Row with Label left-parenthesis 1.1 .8 right-parenthesis EndLabel 1st Column Blank 2nd Column q Superscript h Baseline asterisk m 0 equals q Superscript mathematical left-angle h comma normal upper Lamda mathematical right-angle Baseline m 0 for any h element-of upper P Superscript asterisk Baseline period EndLayout

Then there exists a direct sum decomposition upper M equals circled-plus Underscript lamda element-of upper P Endscripts upper M Subscript lamda (weight space decomposition) where upper M Subscript lamda Baseline equals Overscript normal d normal e normal f period Endscripts left-brace m bar q Superscript h Baseline dot v equals q Superscript mathematical left-angle h comma lamda mathematical right-angle Baseline m for any h element-of upper P Superscript asterisk }. By using the triangular decomposition Equation1.1.6, one can show that the simple highest weight bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis -module is determined uniquely by normal upper Lamda .

We say a bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis -module upper M (defined over double-struck upper Q left-parenthesis q right-parenthesis ) is integrable if upper M has a weight space decomposition upper M equals circled-plus Underscript lamda element-of upper P Endscripts upper M Subscript lamda with dimension upper M Subscript lamda Baseline less-than normal infinity , and for any m element-of upper M , there exists n 0 greater-than-or-equal-to 1 such that e Subscript k Superscript n Baseline asterisk m equals f Subscript k Superscript n Baseline asterisk m equals 0 for all k element-of upper I and n greater-than-or-equal-to n 0 .

The (unique) simple highest weight bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis -module with highest weight normal upper Lamda is integrable if and only if normal upper Lamda is a dominant integral weight normal upper Lamda , i.e., mathematical left-angle normal upper Lamda comma h Subscript k Baseline mathematical right-angle element-of double-struck upper Z Subscript greater-than-or-equal-to 0 for any k element-of upper I (Reference36, 3.5.6, 3.5.8). In this case, the integrable highest weight bold upper U Subscript q -module with highest weight normal upper Lamda is denoted by upper L left-parenthesis normal upper Lamda right-parenthesis .

For a bold upper U Subscript epsilon Baseline left-parenthesis German g right-parenthesis -module upper M (defined over double-struck upper C ), we define highest weight modules, integrable modules, etc. in a similar way.

Suppose normal upper Lamda is dominant. Let upper L left-parenthesis normal upper Lamda right-parenthesis Superscript double-struck upper Z Baseline equals Overscript normal d normal e normal f period Endscripts bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis German g right-parenthesis asterisk m 0 , where m 0 is the highest weight vector. It is known that the natural map upper L left-parenthesis normal upper Lamda right-parenthesis Superscript double-struck upper Z Baseline circled-times Subscript double-struck upper Z left-bracket q comma q Sub Superscript negative 1 Subscript right-bracket Baseline double-struck upper Q left-parenthesis q right-parenthesis right-arrow upper L left-parenthesis normal upper Lamda right-parenthesis is an isomorphism and upper L left-parenthesis normal upper Lamda right-parenthesis Superscript double-struck upper Z circled-times Subscript double-struck upper Z left-bracket q comma q Sub Superscript negative 1 Subscript right-bracket Baseline double-struck upper C is the simple integrable highest weight module of the corresponding Kac-Moody algebra German g with highest weight normal upper Lamda , where double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket right-arrow double-struck upper C is the homomorphism that sends q to 1 (Reference36, Chapter 14 and 33.1.3). Unless epsilon is a root of unity, the simple integrable highest weight bold upper U Subscript epsilon Baseline left-parenthesis German g right-parenthesis -module is the specialization of upper L left-parenthesis normal upper Lamda right-parenthesis Superscript double-struck upper Z (Reference10, 10.1.14, 10.1.15).

1.2. Quantum affine algebra

The quantum affinization bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis of bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis (or simply quantum affine algebra) is an associative algebra over double-struck upper Q left-parenthesis q right-parenthesis generated by e Subscript k comma r Baseline comma f Subscript k comma r Baseline ( k element-of upper I , r element-of double-struck upper Z ), q Superscript h ( h element-of upper P Superscript asterisk ), q Superscript plus-or-minus c slash 2 , q Superscript plus-or-minus d , and h Subscript k comma m ( k element-of upper I , m element-of double-struck upper Z minus StartSet 0 EndSet ) with the following defining relations:

StartLayout 1st Row with Label left-parenthesis 1.2 .1 right-parenthesis EndLabel q Superscript plus-or-minus c slash 2 Baseline is central comma 2nd Row with Label left-parenthesis 1.2 .2 right-parenthesis EndLabel q Superscript 0 Baseline equals 1 comma q Superscript h Baseline q Superscript h Super Superscript prime Superscript Baseline equals q Superscript h plus h Super Superscript prime Superscript Baseline comma left-bracket q Superscript h Baseline comma h Subscript k comma m Baseline right-bracket equals 0 comma q Superscript d Baseline q Superscript negative d Baseline equals 1 comma q Superscript c slash 2 Baseline q Superscript negative c slash 2 Baseline equals 1 comma 3rd Row with Label left-parenthesis 1.2 .3 right-parenthesis EndLabel psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis psi Subscript l Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis equals psi Subscript l Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis comma 4th Row with Label left-parenthesis 1.2 .4 right-parenthesis EndLabel psi Subscript k Superscript minus Baseline left-parenthesis z right-parenthesis psi Subscript l Superscript plus Baseline left-parenthesis w right-parenthesis equals StartFraction left-parenthesis z minus q Superscript minus left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript l Superscript right-parenthesis Baseline q Superscript c Baseline w right-parenthesis left-parenthesis z minus q Superscript left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript l Superscript right-parenthesis Baseline q Superscript negative c Baseline w right-parenthesis Over left-parenthesis z minus q Superscript left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript l Superscript right-parenthesis Baseline q Superscript c Baseline w right-parenthesis left-parenthesis z minus q Superscript minus left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript l Superscript right-parenthesis Baseline q Superscript negative c Baseline w right-parenthesis EndFraction psi Subscript l Superscript plus Baseline left-parenthesis w right-parenthesis psi Subscript k Superscript minus Baseline left-parenthesis z right-parenthesis comma 5th Row with Label left-parenthesis 1.2 .5 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column left-bracket q Superscript d Baseline comma q Superscript h Baseline right-bracket equals 0 comma q Superscript d Baseline h Subscript k comma m Baseline q Superscript negative d Baseline equals q Superscript m Baseline h Subscript k comma m Baseline comma 2nd Row 1st Column Blank 2nd Column q Superscript d Baseline e Subscript k comma r Baseline q Superscript negative d Baseline equals q Superscript r Baseline e Subscript k comma r Baseline comma q Superscript d Baseline f Subscript k comma r Baseline q Superscript negative d Baseline equals q Superscript r Baseline f Subscript k comma r Baseline comma EndLayout 6th Row with Label left-parenthesis 1.2 .6 right-parenthesis EndLabel q Superscript h Baseline e Subscript k comma r Baseline q Superscript negative h Baseline equals q Superscript mathematical left-angle h comma alpha Super Subscript k Superscript mathematical right-angle Baseline e Subscript k comma r Baseline comma q Superscript h Baseline f Subscript k comma r Baseline q Superscript negative h Baseline equals q Superscript minus mathematical left-angle h comma alpha Super Subscript k Superscript mathematical right-angle Baseline f Subscript k comma r Baseline comma 7th Row with Label left-parenthesis 1.2 .7 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column left-parenthesis q Superscript plus-or-minus s c slash 2 Baseline z minus q Superscript plus-or-minus mathematical left-angle h Super Subscript k Superscript comma alpha Super Subscript l Superscript mathematical right-angle Baseline w right-parenthesis psi Subscript l Superscript s Baseline left-parenthesis z right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis 2nd Row 1st Column Blank 2nd Column equals left-parenthesis q Superscript plus-or-minus mathematical left-angle h Super Subscript k Superscript comma alpha Super Subscript l Superscript mathematical right-angle Baseline q Superscript plus-or-minus s c slash 2 Baseline z minus w right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis psi Subscript l Superscript s Baseline left-parenthesis z right-parenthesis comma EndLayout 8th Row with Label left-parenthesis 1.2 .8 right-parenthesis EndLabel left-bracket x Subscript k Superscript plus Baseline left-parenthesis z right-parenthesis comma x Subscript l Superscript minus Baseline left-parenthesis w right-parenthesis right-bracket equals StartFraction delta Subscript k l Baseline Over q Subscript k Baseline minus q Subscript k Superscript negative 1 Baseline EndFraction StartSet delta left-parenthesis q Superscript c Baseline StartFraction w Over z EndFraction right-parenthesis psi Subscript k Superscript plus Baseline left-parenthesis q Superscript c slash 2 Baseline w right-parenthesis minus delta left-parenthesis q Superscript c Baseline StartFraction z Over w EndFraction right-parenthesis psi Subscript k Superscript minus Baseline left-parenthesis q Superscript c slash 2 Baseline z right-parenthesis EndSet comma 9th Row with Label left-parenthesis 1.2 .9 right-parenthesis EndLabel left-parenthesis z minus q Superscript plus-or-minus 2 Baseline w right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis equals left-parenthesis q Superscript plus-or-minus 2 Baseline z minus w right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis comma 10th Row with Label left-parenthesis 1.2 .10 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column product Underscript p equals 1 Overscript minus mathematical left-angle alpha Subscript k Baseline comma h Subscript l Baseline mathematical right-angle Endscripts left-parenthesis z minus q Superscript plus-or-minus left-parenthesis b prime minus 2 p right-parenthesis Baseline w right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis x Subscript l Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis 2nd Row 1st Column Blank 2nd Column equals product Underscript p equals 1 Overscript minus mathematical left-angle alpha Subscript k Baseline comma h Subscript l Baseline mathematical right-angle Endscripts left-parenthesis q Superscript plus-or-minus left-parenthesis b prime minus 2 p right-parenthesis Baseline z minus w right-parenthesis x Subscript l Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis comma if k not-equals l comma EndLayout EndLayout

StartLayout 1st Row with Label left-parenthesis 1.2 .11 right-parenthesis EndLabel sigma-summation Underscript sigma element-of upper S Subscript b Endscripts sigma-summation Underscript p equals 0 Overscript b Endscripts left-parenthesis negative 1 right-parenthesis Superscript p Baseline StartBinomialOrMatrix b Choose p EndBinomialOrMatrix Subscript q Sub Subscript k Baseline x Subscript k Superscript plus-or-minus Baseline left-parenthesis z Subscript sigma left-parenthesis 1 right-parenthesis Baseline right-parenthesis midline-horizontal-ellipsis x Subscript k Superscript plus-or-minus Baseline left-parenthesis z Subscript sigma left-parenthesis p right-parenthesis Baseline right-parenthesis x Subscript l Superscript plus-or-minus Baseline left-parenthesis w right-parenthesis x Subscript k Superscript plus-or-minus Baseline left-parenthesis z Subscript sigma left-parenthesis p plus 1 right-parenthesis Baseline right-parenthesis 2nd Row midline-horizontal-ellipsis x Subscript k Superscript plus-or-minus Baseline left-parenthesis z Subscript sigma left-parenthesis b right-parenthesis Baseline right-parenthesis equals 0 comma if k not-equals l comma EndLayout where q Subscript k Baseline equals q Superscript left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis slash 2 , s equals plus-or-minus , b equals 1 minus mathematical left-angle h Subscript k Baseline comma alpha Subscript l Baseline mathematical right-angle , b prime equals minus left-parenthesis alpha Subscript k Baseline comma alpha Subscript l Baseline right-parenthesis , and upper S Subscript b is the symmetric group of b letters. Here delta left-parenthesis z right-parenthesis , x Subscript k Superscript plus Baseline left-parenthesis z right-parenthesis , x Subscript k Superscript minus Baseline left-parenthesis z right-parenthesis , psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis are generating functions defined by

StartLayout 1st Row delta left-parenthesis z right-parenthesis equals Overscript normal d normal e normal f period Endscripts sigma-summation Underscript r equals negative normal infinity Overscript normal infinity Endscripts z Superscript r Baseline comma x Subscript k Superscript plus Baseline left-parenthesis z right-parenthesis equals Overscript normal d normal e normal f period Endscripts sigma-summation Underscript r equals negative normal infinity Overscript normal infinity Endscripts e Subscript k comma r Baseline z Superscript negative r Baseline comma x Subscript k Superscript minus Baseline left-parenthesis z right-parenthesis equals Overscript normal d normal e normal f period Endscripts sigma-summation Underscript r equals negative normal infinity Overscript normal infinity Endscripts f Subscript k comma r Baseline z Superscript negative r Baseline comma 2nd Row psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis equals Overscript normal d normal e normal f period Endscripts q Superscript plus-or-minus left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis h Super Subscript k Superscript slash 2 Baseline exp left-parenthesis plus-or-minus left-parenthesis q Subscript k Baseline minus q Subscript k Superscript negative 1 Baseline right-parenthesis sigma-summation Underscript m equals 1 Overscript normal infinity Endscripts h Subscript k comma plus-or-minus m Baseline z Superscript minus-or-plus m Baseline right-parenthesis period EndLayout

We will also need the following generating function later:

p Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis equals Overscript normal d normal e normal f period Endscripts exp left-parenthesis minus sigma-summation Underscript m equals 1 Overscript normal infinity Endscripts StartFraction h Subscript k comma plus-or-minus m Baseline Over left-bracket m right-bracket Subscript q Sub Subscript k Subscript Baseline EndFraction z Superscript minus-or-plus m Baseline right-parenthesis period

We have psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis equals q Superscript plus-or-minus left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis h Super Subscript k Superscript slash 2 Baseline p Subscript k Superscript plus-or-minus Baseline left-parenthesis q Subscript k Baseline z right-parenthesis slash p Subscript k Superscript plus-or-minus Baseline left-parenthesis q Subscript k Superscript negative 1 Baseline z right-parenthesis period

Remark 1.2.12

When German g is finite dimensional, then min left-parenthesis mathematical left-angle alpha Subscript k Baseline comma h Subscript l Baseline mathematical right-angle comma mathematical left-angle alpha Subscript l Baseline comma h Subscript k Baseline mathematical right-angle right-parenthesis equals 0 or 1 . 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 German g .

Let bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis Superscript plus (resp. bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis Superscript minus ) be the double-struck upper Q left-parenthesis q right-parenthesis -subalgebra of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis generated by the elements e Subscript k comma r (resp. f Subscript k comma r ). Let bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis Superscript 0 be the double-struck upper Q left-parenthesis q right-parenthesis -subalgebra generated by the elements q Superscript h , h Subscript k comma m .

The quantum loop algebra bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis is the subalgebra of bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis slash left-parenthesis q Superscript plus-or-minus c slash 2 Baseline minus 1 right-parenthesis generated by e Subscript k comma r Baseline comma f Subscript k comma r Baseline ( k element-of upper I , r element-of double-struck upper Z ), q Superscript h ( h element-of upper P Superscript asterisk ), and h Subscript k comma m ( k element-of upper I , m element-of double-struck upper Z minus StartSet 0 EndSet ), i.e., generators other than q Superscript plus-or-minus c slash 2 , q Superscript plus-or-minus d . We will be concerned only with the quantum loop algebra, and not with the quantum affine algebra in the sequel.

There is a homomorphism bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis right-arrow bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis defined by

q Superscript h Baseline right-arrow from bar q Superscript h Baseline comma e Subscript k Baseline right-arrow from bar e Subscript k comma 0 Baseline comma f Subscript k Baseline right-arrow from bar f Subscript k comma 0 Baseline period

Let e Subscript k comma r Superscript left-parenthesis n right-parenthesis Baseline equals Overscript normal d normal e normal f period Endscripts e Subscript k comma r Superscript n Baseline slash left-bracket n right-bracket Subscript q Sub Subscript k Subscript Baseline factorial and f Subscript k comma r Superscript left-parenthesis n right-parenthesis Baseline equals Overscript normal d normal e normal f period Endscripts f Subscript k comma r Superscript n Baseline slash left-bracket n right-bracket Subscript q Sub Subscript k Subscript Baseline factorial . Let bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis be the double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket -subalgebra generated by e Subscript k comma r Superscript left-parenthesis n right-parenthesis , f Subscript k comma r Superscript left-parenthesis n right-parenthesis , q Superscript h and the coefficients of p Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis for k element-of upper I , r element-of double-struck upper Z , n element-of double-struck upper Z Subscript greater-than 0 , h element-of upper P Superscript asterisk . (It should be true that bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis is free over double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket and that the natural map bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis circled-times Subscript double-struck upper Z left-bracket q comma q Sub Superscript negative 1 Subscript right-bracket Baseline double-struck upper Q left-parenthesis q right-parenthesis right-arrow bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis is an isomorphism. But the author does not know how to prove this.) This subalgebra was introduced by Chari-Pressley Reference12. Let bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis Superscript plus (resp. bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis Superscript minus ) be the double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket -subalgebra generated by e Subscript k comma r Superscript left-parenthesis n right-parenthesis (resp. f Subscript k comma r Superscript left-parenthesis n right-parenthesis ) for k element-of upper I , r element-of double-struck upper Z , n element-of upper Z Subscript greater-than 0 . We have bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis Superscript plus-or-minus subset-of bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis . Let bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis Superscript 0 be the double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket -subalgebra generated by q Superscript h , the coefficients of p Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis and

StartBinomialOrMatrix q Superscript h Super Subscript k Superscript Baseline semicolon n Choose r EndBinomialOrMatrix equals Overscript normal d normal e normal f period Endscripts product Underscript s equals 1 Overscript r Endscripts StartFraction q Superscript left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis h Super Subscript k Superscript slash 2 Baseline q Subscript k Superscript n minus s plus 1 Baseline minus q Superscript minus left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis h Super Subscript k Superscript slash 2 Baseline q Subscript k Superscript negative n plus s minus 1 Baseline Over q Subscript k Superscript s Baseline minus q Subscript k Superscript negative s Baseline EndFraction

for all h element-of upper P , k element-of upper I , n element-of double-struck upper Z , r element-of double-struck upper Z Subscript greater-than 0 . One can easily show that bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis Superscript 0 subset-of bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis (see, e.g., Reference36, 3.1.9).

For epsilon element-of double-struck upper C Superscript asterisk , let bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis be the specialized quantum loop algebra defined by bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis circled-times Subscript double-struck upper Z left-bracket q comma q Sub Superscript negative 1 Subscript right-bracket Baseline double-struck upper C via the algebra homomorphism double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket right-arrow double-struck upper C that takes q to epsilon . We assume epsilon is not a root of unity in this paper. Let bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript plus-or-minus and bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript 0 be the specializations of bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis Superscript plus-or-minus and bold upper U Subscript q Superscript double-struck upper Z Baseline left-parenthesis bold upper L German g right-parenthesis Superscript 0 respectively. We have a weak form of the triangular decomposition

StartLayout 1st Row with Label left-parenthesis 1.2 .13 right-parenthesis EndLabel bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis equals bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript minus Baseline dot bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript 0 Baseline dot bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript plus Baseline comma EndLayout

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

We say a bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -module upper M is an l-highest weight module (‘l’ stands for the loop) with l-highest weight left-parenthesis normal upper Lamda comma left-parenthesis normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis right-parenthesis Subscript k Baseline right-parenthesis (where normal upper Lamda element-of upper P , left-parenthesis normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis right-parenthesis Subscript k Baseline element-of double-struck upper C left-bracket left-bracket z Superscript minus-or-plus Baseline right-bracket right-bracket Superscript upper I ) if there exists a vector m 0 element-of upper M such that

StartLayout 1st Row with Label left-parenthesis 1.2 .14 right-parenthesis EndLabel 1st Column Blank 2nd Column e Subscript k comma r Baseline asterisk m 0 equals 0 comma bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript minus Baseline asterisk m 0 equals upper M comma 2nd Row with Label left-parenthesis 1.2 .15 right-parenthesis EndLabel 1st Column Blank 2nd Column q Superscript h Baseline asterisk m 0 equals epsilon Superscript mathematical left-angle h comma normal upper Lamda mathematical right-angle Baseline m 0 for h element-of upper P Superscript asterisk Baseline comma psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis asterisk m 0 equals normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis m 0 for k element-of upper I period EndLayout

By using Equation1.2.13 and a standard argument, one can show that there is a simple l-highest weight module upper M of bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis with l-highest weight vector m 0 satisfying the above for any left-parenthesis normal upper Lamda comma left-parenthesis normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis right-parenthesis Subscript k Baseline right-parenthesis with normal upper Psi Subscript k Superscript plus Baseline left-parenthesis normal infinity right-parenthesis equals left-parenthesis alpha Subscript k Baseline comma alpha Subscript k Baseline right-parenthesis mathematical left-angle normal upper Lamda comma h Subscript k Baseline mathematical right-angle slash 2 , normal upper Psi Subscript k Superscript minus Baseline left-parenthesis 0 right-parenthesis equals minus left-parenthesis alpha Subscript k Baseline comma alpha Subscript k Baseline right-parenthesis mathematical left-angle normal upper Lamda comma h Subscript k Baseline mathematical right-angle slash 2 . Moreover, such upper M is unique up to isomorphism. For abuse of notation, we denote the pair left-parenthesis normal upper Lamda comma left-parenthesis normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis right-parenthesis Subscript k Baseline right-parenthesis simply by the symbol normal upper Psi Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis .

A bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -module upper M is said to be l-integrable if

(a)

upper M has a weight space decomposition upper M equals circled-plus Underscript lamda element-of upper P Endscripts upper M Subscript lamda as a bold upper U Subscript epsilon Baseline left-parenthesis German g right-parenthesis -module such that dimension upper M Subscript lamda Baseline less-than normal infinity ,

(b)

for any m element-of upper M , there exists n 0 greater-than-or-equal-to 1 such that e Subscript k comma r 1 Baseline midline-horizontal-ellipsis e Subscript k comma r Sub Subscript n Subscript Baseline asterisk m equals f Subscript k comma r 1 Baseline midline-horizontal-ellipsis f Subscript k comma r Sub Subscript n Subscript asterisk m equals 0 for all r 1 comma ellipsis comma r Subscript n Baseline element-of double-struck upper Z , k element-of upper I and n greater-than-or-equal-to n 0 .

For example, if German g is finite dimensional, and upper M is a finite dimensional module, then upper M satisfies the above conditions after twisting with a certain automorphism of bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis (Reference10, 12.2.3).

Proposition 1.2.16

Assume that German g is symmetric. The simple l-highest weight bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -module upper M with l-highest weight normal upper Psi Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis is l-integrable if and only if normal upper Lamda is dominant and there exist polynomials upper P Subscript k Baseline left-parenthesis u right-parenthesis element-of double-struck upper C left-bracket u right-bracket for k element-of upper I with upper P Subscript k Baseline left-parenthesis 0 right-parenthesis equals 1 such that

StartLayout 1st Row with Label left-parenthesis 1.2 .17 right-parenthesis EndLabel normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis equals epsilon Subscript k Superscript degree upper P Super Subscript k Superscript Baseline left-parenthesis StartFraction upper P Subscript k Baseline left-parenthesis epsilon Subscript k Superscript negative 1 Baseline slash z right-parenthesis Over upper P Subscript k Baseline left-parenthesis epsilon Subscript k Baseline slash z right-parenthesis EndFraction right-parenthesis Superscript plus-or-minus Baseline comma EndLayout

where epsilon Subscript k Baseline equals epsilon Superscript left-parenthesis alpha Super Subscript k Superscript comma alpha Super Subscript k Superscript right-parenthesis slash 2 , and left-parenthesis right-parenthesis Superscript plus-or-minus Baseline element-of double-struck upper C left-bracket left-bracket z Superscript minus-or-plus Baseline right-bracket right-bracket denotes the expansion at z equals normal infinity and 0 respectively.

This result was announced by Drinfel’d for the Yangian Reference15. The proof of the ‘only if’ part when German g is finite dimensional was given by Chari-Pressley Reference10, 12.2.6. Since the proof is based on a reduction to the case German g equals German s German l Subscript 2 , it can be applied to a general Kac-Moody algebra German g (not necessarily symmetric). The ‘if’ part was proved by them later in Reference11 when German g 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 German g in §13. Our proof is independent of Chari-Pressley’s proof.

Remark 1.2.18

The polynomials upper P Subscript k are called Drinfel’d polynomials.

When the Drinfel’d polynomials are given by

upper P Subscript k Baseline left-parenthesis u right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 minus s u 2nd Column if k not-equals k 0 comma 2nd Row 1st Column 1 2nd Column otherwise comma EndLayout

for some k 0 element-of upper I , s element-of double-struck upper C Superscript asterisk , the corresponding simple l-highest weight module is called an l-fundamental representation. When German g is finite dimensional, bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis is a Hopf algebra since Drinfel’d Reference15 announced and Beck Reference5 proved that bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis can be identified with (a quotient of) the specialized quantized enveloping algebra associated with Cartan data of affine type. Thus a tensor product of bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -modules is again a bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -module. We have the following:

Proposition 1.2.19 (Reference10, 12.2.6,12.2.8).

Suppose German g is finite dimensional.

(1) If upper M and upper N are simple l-highest weight bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -modules with Drinfel’d polynomials upper P Subscript k comma upper M , upper P Subscript k comma upper N such that upper M circled-times upper N is simple, then its Drinfel’d polynomial upper P Subscript k comma upper M circled-times upper N is given by

upper P Subscript k comma upper M circled-times upper N Baseline equals upper P Subscript k comma upper M Baseline upper P Subscript k comma upper N Baseline period

(2) Every simple l-highest weight bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -module is a subquotient of a tensor product of l-fundamental representations.

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

1.3. An l-weight space decomposition

Let upper M be an l-integrable bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis -module with the weight space decomposition upper M equals circled-plus Underscript lamda element-of upper P Endscripts upper M Subscript lamda . Since the commutative subalgebra bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript 0 preserves each upper M Subscript lamda , we can further decompose upper M into a sum of generalized simultaneous eigenspaces for bold upper U Subscript epsilon Baseline left-parenthesis bold upper L German g right-parenthesis Superscript 0 :

StartLayout 1st Row with Label left-parenthesis 1.3 .1 right-parenthesis EndLabel upper M equals circled-plus upper M Subscript normal upper Psi Sub Superscript plus-or-minus Subscript Baseline comma EndLayout

where normal upper Psi Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis is a pair left-parenthesis normal upper Lamda comma left-parenthesis normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis right-parenthesis Subscript k Baseline right-parenthesis as before and

upper M Subscript normal upper Psi Sub Superscript plus-or-minus Subscript Baseline equals Overscript normal d normal e normal f period Endscripts left-brace m element-of upper M vertical-bar StartLayout 1st Row 1st Column Blank 2nd Column q Superscript h Baseline asterisk m equals epsilon Superscript mathematical left-angle h comma normal upper Lamda mathematical right-angle Baseline m for h element-of upper P Superscript asterisk Baseline comma 2nd Row 1st Column Blank 2nd Column left-parenthesis psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis minus normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis upper I d right-parenthesis Superscript upper N Baseline asterisk m equals 0 3rd Row 1st Column Blank 2nd Column for k element-of upper I and sufficiently large upper N EndLayout right-brace period

If upper M Subscript normal upper Psi Sub Superscript plus-or-minus Baseline not-equals 0 , we call upper M Subscript normal upper Psi Sub Superscript plus-or-minus an l-weight space, and the corresponding normal upper Psi Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis 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 normal upper Psi Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis equals left-parenthesis normal upper Lamda comma left-parenthesis normal upper Psi Subscript k Superscript plus-or-minus Baseline left-parenthesis z right-parenthesis right-parenthesis Subscript k Baseline right-parenthesis is said to be l-dominant if normal upper Lamda is dominant and there exists a polynomial upper P left-parenthesis u right-parenthesis equals left-parenthesis upper P Subscript k Baseline left-parenthesis u right-parenthesis right-parenthesis Subscript k Baseline element-of double-struck upper C left-bracket u right-bracket Superscript upper I for with upper P Subscript k Baseline left-parenthesis 0 right-parenthesis equals 1 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 upper I be the set of vertices and upper E the set of edges. Let bold upper A be the adjacency matrix of the graph, namely

bold upper A equals left-parenthesis bold upper A Subscript k l Baseline right-parenthesis Subscript k comma l element-of upper I Baseline comma where bold upper A Subscript k l Baseline is the number of edges joining k and l period

We associate with the graph left-parenthesis upper I comma upper E right-parenthesis a symmetric generalized Cartan matrix bold upper C equals 2 bold upper I minus bold upper A , where bold upper I 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 German g , the quantized enveloping algebra bold upper U Subscript q Baseline left-parenthesis German g right-parenthesis , the quantum affine algebra bold upper U Subscript q Baseline left-parenthesis ModifyingAbove German g With caret right-parenthesis and the quantum loop algebra bold upper U Subscript q Baseline left-parenthesis bold upper L German g right-parenthesis . Let upper H be the set of pairs consisting of an edge together with its orientation. For h element-of upper H , we denote by i n left-parenthesis h right-parenthesis (resp. o u t left-parenthesis h right-parenthesis ) the incoming (resp. outgoing) vertex of h . For h element-of upper H we denote by h overbar the same edge as h with the reverse orientation. Choose and fix an orientation normal upper Omega of the graph, i.e., a subset normal upper Omega subset-of upper H such that normal upper Omega overbar union normal upper Omega equals upper H , normal upper Omega intersection normal upper Omega overbar equals normal empty-set . The pair left-parenthesis upper I comma normal upper Omega right-parenthesis is called a quiver. Let us define matrices bold upper A Subscript normal upper Omega and bold upper A Subscript normal upper Omega overbar by

StartLayout 1st Row with Label left-parenthesis 2.1 .1 right-parenthesis EndLabel StartLayout 1st Row 1st Column left-parenthesis bold upper A Subscript normal upper Omega Baseline right-parenthesis Subscript k l 2nd Column equals Overscript normal d normal e normal f period Endscripts number-sign StartSet h element-of normal upper Omega bar i n left-parenthesis h right-parenthesis equals k comma o u t left-parenthesis h right-parenthesis equals l EndSet comma 2nd Row 1st Column left-parenthesis bold upper A Subscript normal upper Omega overbar Baseline right-parenthesis Subscript k l 2nd Column equals Overscript normal d normal e normal f period Endscripts number-sign StartSet h element-of normal upper Omega overbar bar i n left-parenthesis h right-parenthesis equals k comma o u t left-parenthesis h right-parenthesis equals l EndSet period EndLayout EndLayout

So we have bold upper A equals bold upper A Subscript normal upper Omega Baseline plus bold upper A Subscript normal upper Omega overbar , Superscript t Baseline bold upper A Subscript normal upper Omega Baseline equals bold upper A Subscript normal upper Omega overbar .

Let upper V equals left-parenthesis upper V Subscript k Baseline right-parenthesis Subscript k element-of upper I be a collection of finite-dimensional vector spaces over double-struck upper C for each vertex k element-of upper I . The dimension of upper V is a vector

dimension upper V equals left-parenthesis dimension upper V Subscript k Baseline right-parenthesis Subscript k element-of upper I Baseline element-of double-struck upper Z Subscript greater-than-or-equal-to 0 Superscript upper I Baseline period

If upper V Superscript 1 and upper V squared are such collections, we define vector spaces by

StartLayout 1st Row with Label left-parenthesis 2.1 .2 right-parenthesis EndLabel StartLayout 1st Row 1st Column Blank 2nd Column normal upper L left-parenthesis upper V Superscript 1 Baseline comma upper V squared right-parenthesis equals Overscript normal d normal e normal f period Endscripts circled-plus Underscript k element-of upper I Endscripts upper H o m left-parenthesis upper V Subscript k Superscript 1 Baseline comma upper V Subscript k Superscript 2 Baseline right-parenthesis comma 2nd Row 1st Column Blank 2nd Column normal upper E left-parenthesis upper V Superscript 1 Baseline comma upper V squared right-parenthesis equals Overscript normal d normal e normal f period Endscripts circled-plus Underscript h element-of upper H Endscripts upper H o m left-parenthesis upper V Subscript o u t left-parenthesis h right-parenthesis Superscript 1 Baseline comma upper V Subscript i n left-parenthesis h right-parenthesis Superscript 2 Baseline right-parenthesis period EndLayout EndLayout

For upper B equals left-parenthesis upper B Subscript h Baseline right-parenthesis element-of normal upper E left-parenthesis upper V Superscript 1 Baseline comma upper V squared right-parenthesis and upper C equals left-parenthesis upper C Subscript h Baseline right-parenthesis element-of normal upper E left-parenthesis upper V squared comma upper V cubed right-parenthesis , let us define a multiplication of upper B and upper C by

upper C upper B equals Overscript normal d normal e normal f period Endscripts left-parenthesis sigma-summation Underscript i n left-parenthesis h right-parenthesis equals k Endscripts upper C Subscript h Baseline upper B Subscript h overbar Baseline right-parenthesis Subscript k Baseline element-of normal upper L left-parenthesis upper V Superscript 1 Baseline comma upper V cubed right-parenthesis period

Multiplications b a , upper B a of a element-of normal upper L left-parenthesis upper V Superscript 1 Baseline comma upper V squared right-parenthesis , b element-of normal upper L left-parenthesis upper V squared comma upper V cubed right-parenthesis , upper B element-of normal upper E left-parenthesis upper V squared comma upper V cubed right-parenthesis are defined in an obvious manner. If a element-of normal upper L left-parenthesis upper V Superscript 1 Baseline comma upper V Superscript 1 Baseline right-parenthesis , its trace trace left-parenthesis a right-parenthesis is understood as sigma-summation Underscript k Endscripts trace left-parenthesis a Subscript k Baseline right-parenthesis .

For two collections upper V , upper W of vector spaces with bold v equals dimension upper V , bold w equals dimension upper W , we consider the vector space given by

StartLayout 1st Row with Label left-parenthesis 2.1 .3 right-parenthesis EndLabel bold upper M identical-to bold upper M left-parenthesis bold v comma bold w right-parenthesis equals Overscript normal d normal e normal f period Endscripts normal upper E left-parenthesis upper V comma upper V right-parenthesis circled-plus normal upper L left-parenthesis upper W comma upper V right-parenthesis circled-plus normal upper L left-parenthesis upper V comma upper W right-parenthesis comma EndLayout

where we use the notation bold upper M unless we want to specify dimensions of upper V , upper W . The above three components for an element of bold upper M will be denoted by upper B , i , j respectively. An element of bold upper M will be called an ADHM datum.

Usually a point in circled-plus Underscript h element-of normal upper Omega Endscripts upper H o m left-parenthesis upper V Subscript o u t left-parenthesis h right-parenthesis Superscript 1 Baseline comma upper V Subscript i n left-parenthesis h right-parenthesis Superscript 2 Baseline right-parenthesis is called a representation of the quiver left-parenthesis upper I comma normal upper Omega right-parenthesis in the literature. Thus normal upper E left-parenthesis upper V comma upper V right-parenthesis is the product of the space of representations of left-parenthesis upper I comma normal upper Omega right-parenthesis and that of left-parenthesis upper I comma normal upper Omega overbar right-parenthesis . On the other hand, the factor normal upper L left-parenthesis upper W comma upper V right-parenthesis or normal upper L left-parenthesis upper V comma upper W right-parenthesis 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:

bold v right-arrow from bar sigma-summation Underscript k Endscripts v Subscript k Baseline alpha Subscript k Baseline comma bold w right-arrow from bar sigma-summation Underscript k Endscripts w Subscript k Baseline normal upper Lamda Subscript k Baseline comma

where v Subscript k (resp. w Subscript k ) is the k th component of bold v (resp. bold w ). Since StartSet alpha Subscript k Baseline EndSet and StartSet normal upper Lamda Subscript k Baseline EndSet are both linearly independent, these maps are injective. We consider bold v and bold w as elements of the weight lattice upper P in this way hereafter.

For a collection upper S equals left-parenthesis upper S Subscript k Baseline right-parenthesis Subscript k element-of upper I of subspaces of upper V Subscript k and upper B element-of normal upper E left-parenthesis upper V comma upper V right-parenthesis , we say upper S is upper B -invariant if upper B Subscript h Baseline left-parenthesis upper S Subscript o u t left-parenthesis h right-parenthesis Baseline right-parenthesis subset-of upper S Subscript i n left-parenthesis h right-parenthesis .

Fix a function epsilon colon upper H right-arrow double-struck upper C Superscript asterisk such that epsilon left-parenthesis h right-parenthesis plus epsilon left-parenthesis h overbar right-parenthesis equals 0 for all h element-of upper H . In Reference44Reference45, it was assumed that epsilon takes its value plus-or-minus 1 , but this assumption is not necessary as remarked by Lusztig Reference38. For upper B element-of normal upper E left-parenthesis upper V Superscript 1 Baseline comma upper V squared right-parenthesis , let us denote by epsilon upper B element-of normal upper E left-parenthesis upper V Superscript 1 Baseline comma upper V squared right-parenthesis data given by left-parenthesis epsilon upper B right-parenthesis Subscript h Baseline equals epsilon left-parenthesis h right-parenthesis upper B Subscript h for h element-of upper H .

Let us define a symplectic form omega on bold upper M by

StartLayout 1st Row with Label left-parenthesis 2.1 .5 right-parenthesis EndLabel omega left-parenthesis left-parenthesis upper B comma i comma j right-parenthesis comma left-parenthesis upper B prime comma i prime comma j Superscript prime Baseline right-parenthesis right-parenthesis equals Overscript normal d normal e normal f period Endscripts trace left-parenthesis epsilon upper B upper B Superscript prime Baseline right-parenthesis plus trace left-parenthesis i j Superscript prime Baseline minus i prime j right-parenthesis period EndLayout