Schedule

21COE, RIMS Research Project 2004

Method of Algebraic Analysis in Integrable Systems

Representation Theory and Geometry

Aug. 12 (Thu.) -- 13 (Fri.), 2004

Kyoto University

Department of Mathematics, Sci.Bldg.No.6 (New Bldg.), Room 609

Program

 8/12 (Thu.) 13:30 -- 14:30 R.Bezrukavnikov (Northwestern) Derived categories of symplectic resolutions via quantization in positive characteristic 14:45 -- 15:45 A.Braverman (Harvard) Seiberg-Witten prepotential for general Schrodinger operators and geometric applications 16:00 -- 17:00 M.Finkelberg (Moscow) Equivariant homology of affine Grassmannian and Toda lattice 8/13 (Fri.) 10:00 -- 11:00 E.Frenkel (Berkeley) Local Langlands correspondence for affine Kac-Moody algebras 11:15 -- 12:15 D.Hernandez (ENS) A new fusion procedure for affinizations of quantum Kac-Moody algebras 13:30 -- 14:30 S.Loktev (Kyoto) Weyl modules over multi-dimensional currents 14:45 -- 15:45 H.Nakajima (Kyoto) Cells of quantum affine algebras

## Abstract

R.Bezrukavnikov

Title : Derived categories of symplectic resolutions via quantization in positive characteristic

Abstract : I will discuss a method for description of the derived category of coherent sheaves on a (n algebraic) symplectic resolution of singularities. The method originated in the study of representation theory of simple Lie algebras in positive characteristic (joint with Mirkovic and Rumynin), and was applied to the case of resolution of quotient singularities to obtain a McKay type equivalence (joint with Kaledin). Conjecturally it applies to any symplectic resolution, in particular to the quiver varieties

A.Braverman

Title : Seiberg-Witten prepotential for general Schrodinger operators and geometric applications

Abstract : We introduce the notion of a "Seiberg-Witten prepotential" for a very general class of Schrodinger operators with periodic potential (the definition is completely algebraic). In the case when the operator in question is integrable it coincides with the prepotential as defined before by physiscists. As an application we shall sketch the proof of Nekrasov's conjecture which gives a connection between "instanton counting" for a semi-simple group $G$ (with Lie algebra $\mathfrak g$) and the prepotential of the Toda system associated with the affine Lie algebra whose Dynkin diagram is dual to that of the affinisation of $\mathfrak g$. No previous familiarity with the subject will be assumed in the talk.

M.Finkelberg

Title : Equivariant homology of affine Grassmannian and Toda lattice

Abstract : For an affine Grassmannian $Gr_G=G((t))/G[[t]]$, its homology equivariant with respect to $G[[t]]\times C^*$ (semidirect product with loop rotations) form a convolution ring, which is isomorphic to a completion of the quantum Toda lattice for the Langlands dual group $\check G$. It is a quantization of the commutative convolution ring $H_{G[[t]]}(Gr_G)$ which is isomorphic to a completion of the classical Toda lattice. The commuting hamiltonians come from the equivariant homology of the point $H_{G[[t]]}(pt)$. This convolution ring acts on the equivariant homology of the local semiinfinite flag space. This gives rise to the genuine Toda lattice, and Givental's quantum D-module structure on the quantum cohomology of $G/B$. The similar results hold for the K-theory.

E.Frenkel

Title : Local Langlands correspondence for affine Kac-Moody algebras

Abstract : The local Langlands correspondence relates irreducible representations of the group $GL_n(F)$ and $n$-dimensional representations of the Galois group of the field $F=F_q((t))$. More generally, representations of $G(F)$, where $G$ is a reducive algebraic group over $F$ are related to homomorphisms from the Galois group of $F$ to the Langlands dual group of $G$. It is natural to ask what happens if we replace the finite field $F_q$ by the complex field $C$. It turns out that there is indeed an analogue of the local Langlands correspondence for the loop groups, but that one should consider their representations on categories, rather than vector spaces. These categories may be described either as categories of representations of affine Kac-Moody algebras (the central extensions of loop Lie algebras) or as categories of D-modules on the flag varieties of loop groups. This is a joint work with D. Gaitsgory.

D.Hernandez

Title : A new fusion procedure for affinizations of quantum Kac-Moody algebras

Abstract : The class of affinizations of quantum Kac-Moody algebras includes in particular quantum affine algebras and quantum toroidal algebras. In general they have no Hopf algebra structure. However we propose a construction of a fusion product on the Grothendieck group of l-highest weight integrable representations. This new fusion procedure uses a one parameter deformation of the "new Drinfel'd coproduct" and is closely related to a generalization of Frenkel-Reshetikhin q-characters. Moreover in the case of quantum affine algebras it gives a new interpretation of the usual Grothendieck ring.

Sergei Loktev

Title : Weyl modules over multi-dimensional currents

Abstract : Talk is based on the papers math.QA/0212001 and math.QA/0312158 by Boris Feigin and the speaker.

Let $\mathfrak g$ be a simple Lie algebra. Choose a Cartan and a Borel subalgebra $\mathfrak h \subset \mathfrak b \subset \mathfrak g$. Let $A$ be an associative algebra with unit. To avoid technical assumptions let us suppose that $A$ is the algebra of functions on an affine variety $M$ (possibly singular).

The talk is about representations of $\mathfrak g \otimes A$ belonging to the following class. Let $\lambda : \mathfrak b \to \mathfrak h \to \mathbb C$ be a weight, let $\epsilon : A \to \mathbb C$ be an augmentation of $A$ (that is evaluation at a certain point on $M$).

Definition.

The Weyl module $W(\lambda, \epsilon)$ is the maximal finite-dimensional module generated by the vector $v_{\lambda, \epsilon}$ such that

$(\mathfrak g \otimes P) \cdot v_{\lambda, \epsilon} = \lambda(g) \epsilon(P) v_{\lambda, \epsilon},$

where $g \in \mathfrak b$, $P \in A$.

In particular, if $M$ is a point, so $A=1$, then it is just the highest weight representation of $\mathfrak g$.

It can be shown that $W(\lambda, \epsilon)$ exists and that $W(\lambda, \epsilon) \neq 0$ if and only if $\lambda$ is dominant.

First we discuss the case when $\mathfrak g = sl_r$ and $\lambda = n \omega$, where $\omega$ is the weight of the vector representations. Then we show that $W(n \omega,\epsilon)$ as a module over $\mathfrak g \otimes 1$ is related by the Frobenius transformation to the space of diagonal coinvariants

$A^{\otimes n} / \left(S^n(A)_\epsilon \cdot A^{\otimes n}\right),$

where $S^n(A)_\epsilon$ is the augmentation ideal.

The case of $M = \mathbb C$ (and, actually, any non-singular curve) was studied by V.Chari and A.Pressley. Here $\dim W(n \omega,\epsilon) = r^n$. The case of a curve with a double point was recently completed by T.Kuwabara.

We discuss the computation for $M = \mathbb C^2$ (and, actually, any non-singular surface). It is performed using the results of M.Haiman on diagonal coinvariants for this case. The answer is

$\dim W(n \omega,\epsilon) = \frac{(r(n+1))!}{(n+1)!((r-1)(n+1)+1)!},$

that is the higher Catalan numbers (usual ones for $r=2$).

Recall that M.Haiman described the structure of diagonal coinvariants in terms of parking functions. We discuss a deformation of the Weyl modules (together with the Lie algebra $\mathfrak g \otimes A$) into modules with basis related to parking functions.

If there will be enough time we apply this construction to obtain for $M=\mathbb C^2$

1) a simple proof that dimension of the space of diagonal coinvariants is not less than the number of parking functions;

2) a lower bound (conjecturally exact) for dimensions of Weyl modules over $sl_r$ with arbitrary weight $\lambda$;

3) a description of the limits of Weyl modules in terms of semi-infinite forms and the limits of characters.

Hiraku Nakajima

Title : Cells of quantum affine algebras

Abstract : (joint work with J.Beck) We determine two-sided cells and the limit algebra of a quantum affine algebra. The proof is based on study of extremal weight modules, introduced by Kashiwara. When the affine Lie algebra is symmetric, modules are isomorphic to the universal standard modules introduced by the speaker via quiver varieties.

Contact H.Nakajima (nakajima@math.kyoto-u.ac.jp) for any question.