## EVENT DETAILS AND ABSTRACT

**Algebra Seminar**
**Title:** Equivariant (K-)homology of affine Grassmannian and Toda lattice

**Speaker:** Professor Michael Finkelberg

**Speaker Info:** Independent Moscow University

**Brief Description:**

**Special Note**:

**Abstract:**

Let G be a semi-simple compact Lie group with complexification G_c;
the group \omega G of based polynomial loops into G is also known
as the affine Grassmannian Gr_{G_c} of G_c. Certain topological
invariants of \omega G are studied in the geometric Langlands
duality theory, and are related to the Langlands dual group \check G_c.
We describe a new (algebro-geometric) invariant of this kind:
the monoidal category of equivariant perverse coherent sheaves
on Gr_{G_c}. We compute its Grothendieck ring K^G(Gr_{G_c}),
and an 'additive' analogue of the latter, the equivariant homology
H^G(\omega G) ('equivariant loop homology of G'). We identify
H^G(\omega G) with the completed Kostant Toda lattice for \check G_c.
Time permitting, we will discuss conjectures relating these
results to a 'quantization' of the Beilinson-Drinfeld geometric
Langlands conjecture, and also a classical limit of this conjecture;
and relation to Feigin-Loktev fusion product of G_c- modules.

**Date:** Tuesday, December 9, 2003

**Time:** 4:10pm

**Where:** Lunt 104

**Contact Person:** Prof. Thomas Wenger

**Contact email:** wengerth@math.northwestern.edu

**Contact Phone:** 847-491-5544

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