Math Seminars.
Dennis Gaitsgory, (IAS)
Geometric Construction of Central Elements in the Affine Hecke Algebra
Let G be a p-adic group and let H_I denote the
Iwahori Hecke algebra of G, i.e. H_I consists of compactly
supported functions on G, which are bi-invariant with respect to the Iwahori
subgroup of G.
It is well-known that the center of H_I is isomorphic to the
spherical Hecke algebra H_s of G, and the latter can be
identified by means of the Satake isomorphism with the Grothendieck ring of
finite-dimensional representations of the corresponding Langlands' dual group.
Moreover, D. Kazhdan and G. Lusztig gave a geometric interpretation of
H_I. They showed that elements of H_I can be viewed
as perverse sheaves on a certain infinite-dimensional algebraic variety, called
the affine flag manifold.
Therefore, it becomes a natural question to ask if one can construct explicitly
the center of H_I using algebraic geometry. (In this form the
question was first formulated by R. Kottwitz).
In this talk we shall present such a construction, using Drinfeld's idea of
fusion of Hecke operators. Moreover, we shall see that it has many favorable
properties. In particular, it has been used by
R. Bezrukavnikov for the proof of Lusztig's conjecture about the relation
between two-sided cells in the affine Weyl group and centralizers of unipotent
elements in the Langlands' dual group.
Results which are going to be mentioned in this talk were obtained in a joint
work with A. Beilinson.