Compatibility between Satake and Bernstein-type isomorphisms in characteristic p

الأصل · EN

We study the center of the pro-p Iwahori-Hecke ring H of a connected split p-adic reductive group G. For k an algebraically closed field with characteristic p, we prove that the center of the k-algebra Hₖ:= H⊗Z k contains an affine semigroup algebra which is naturally isomorphic to the Hecke algebra attached to any irreducible smooth k-representation of a given hyperspecial maximal compact subgroup of G. This isomorphism is obtained using the inverse Satake isomorphism constructed in arXiv:1207.5557. We apply this to classify the simple supersingular Hₖ-modules, study the supersingular block in the category of finite length Hₖ-modules, and relate the latter to supersingular representations of G.