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Séminaire de Mathématique
# Modular Iwahori-Hecke algebras: A survey and some computations

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

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Avec le soutien de :
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ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

*PI : Michael HARRIS*

`The smooth representation theory of a p-adic reductive group G with characteristic zero coefficients is very closely connected to the module theory of its (pro-p) Iwahori-Hecke algebra H = H(G). In the modular case,where the coefficients have characteristic p, this connection breaks down to a large extent. In this talk I will first survey joint work with R. Ollivier in this modular case. `

We determine completely the homological properties of H, and we introduce a certain torsion theory in the module category Mod(H) such that the torsion free modules embed fully faithfully into the category of smooth G-representations. In the case of the group SL_2 we are able to explicitly compute this torsion theory. Secondly I will describe a derived picture of the whole situation in which one recovers an equivalence between the module theory of a derived version of H and the derived representation theory of G. In both approaches the cohomology of the pro-p Iwahori subgroup of G in a certain universal module plays a crucial role.

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