Let F be a locally compact non-Archimedean field, p its residue characteristic and G a connected reductive algebraic group over F . The classical Satake isomorphism describes the Hecke algebra (over the field of complex numbers) of double classes in G with respect to a special maximal compact subgroup K of G . In our setting K is a slightly smaller special parahoric subgroup, we introduce an absolutely irreducible smooth representation of K on a vector field V over a field of characteristic p , and we get a description of an analoguous Hecke algebra with respect to V . We thus generalize work of F. Herzig treating the case of unramified groups G .

But in our general situation the Hecke algebra can be non-commutative. (This is joint work with M.-F. VignÃ©ras.)

# A Satake Isomorphism $mod.p$

Guy Henniart

University of Paris-Sud

November 4, 2010