Iwahori-Hecke algebras are Gorenstein (part I)

Peter Schneider
Wed, Oct 17, 2012
PIMS, University of British Columbia
PIMS Speaker series

N.B. Due to problems with our camera, there is some audio distortion on this file and a small portion of the video has been removed.

This lecture is the first of a two part series (part II).

In the local Langlands program the (smooth) representation theory
of p-adic reductive groups G in characteristic zero plays a key role. For any
compact open subgroup K of G there is a so called Hecke algebra H(G,K). The
representation theory of G is equivalent to the module theories over all these
algebras H(G,K). Very important examples of such subgroups K are the Iwahori
subgroup and the pro-p Iwahori subgroup. By a theorem of Bernstein the Hecke
algebras of these subgroups (and many others) have finite global dimension.

In recent years the same representation theory of G but over
an algebraically closed field of characteristic p has become more and more important. But little is known yet. Again one can define analogous Hecke algebras. Their relation to the representation theory of G is still very mysterious. Moreover
they are no longer of finite global dimension. In joint work with R. Ollivier
we prove that over any field the algebra H(G,K), for K the (pro-p) Iwahori
subgroup, is Gorenstein.