Compatibility between Satake and Bernstein-type isomorphisms in characteristic p

TL;DR

This paper investigates the center of the pro-p Iwahori-Hecke algebra over a field of characteristic p, establishing an isomorphism with an affine semigroup algebra and applying it to classify supersingular modules and relate them to supersingular representations of the group.

Contribution

It demonstrates the compatibility between Satake and Bernstein-type isomorphisms in characteristic p and uses this to classify supersingular modules and connect them to group representations.

Findings

The center contains an affine semigroup algebra isomorphic to a Hecke algebra.

Classification of simple supersingular modules over the algebra.

Relation established between supersingular modules and supersingular group representations.

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.