Compatibility between Satake and Bernstein-type isomorphisms in characteristic p

Abstract

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 H_k:= H\otimes_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 H_k-modules, study the supersingular block in the category of finite length H_k-modules, and relate the latter to supersingular representations of G.

The new version contains the classification of the simple supersingular Hecke modules in the case of a general split group

References in corpus (1)

• On a classification of irreducible admissible modulo $p$ representations of a $p$-adic split reductive group

Cited by in corpus (3)

• Central elements in affine mod $p$ Hecke algebras via perverse ${\mathbb{F}}_p$-sheaves
• Inverse Satake isomorphism and change of weight
• From pro-$p$ Iwahori-Hecke modules to $(\phi ,\Gamma )$-modules, II