# Iwahori-Hecke algebras are Gorenstein (part I)

**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.

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