Derived Hecke algebras

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(G). In the modular

case, where the coefficients have characteristic p, this connection

breaks down to a large extent. I will first explain how this connection

can be reinstated by passing to a derived setting. It involves a certain

differential graded algebra whose zeroth cohomology is H(G). Then I will

report on a joint project with

R. Ollivier in which we analyze the higher cohomology groups of this dg

algebra for the group G = SL_2.