Harvard-MIT Algebraic Geometry Seminar
		    February 26, 2002 at 3:00 p.m.
			    MIT Room 4-163



      Geometric Langlands correspondence and Kac-Moody algebras


		      Edward Frenkel (Berkeley)



Abstract:

Representation theory of affine Kac-Moody algebras is the source of
some interesting geometry related to local systems on algebraic curves
(known as opers) and moduli spaces of G-bundles on curves.  I will
talk about a conjectural description (joint with D.Gaitsgory) of a
certain category of representations of an affine Kac-Moody algebra in
terms of the (derived) category of quasi-coherent sheaves on the
Springer fiber corresponding to a nilpotent element in the Lie algebra
of the Langlands dual group of G.

This is closely related to the geometric Langlands correspondence.
Given a local system E for the Langlands group of G on a smooth
projective curve X (possibly ramified at some points), one wishes to
describe the category of Hecke eigensheaves with "eigenvalue" E on the
moduli stack of G-bundles on X (with level structures at those
points).  A.Beilinson and V.Drinfeld have shown how to construct Hecke
eigensheaves starting from representations of the affine Kac-Moody
algebra g^ associated to G, so that the categories of Hecke
eigensheaves are closely related to categories of modules over g^.

On the other hand, G.Lusztig has previously defined certain bases in
the equivariant K-theory of the Springer fibers, and we expect that
they correspond to the bases of irreducible g^-modules.