An inverse Satake isomorphism in characteristic p

TL;DR

This paper constructs an inverse Satake isomorphism in characteristic p for split reductive groups over local fields, linking Hecke algebras to dominant cocharacters, and proves its equivalence to Herzig's Satake isomorphism in certain cases.

Contribution

It introduces a new inverse Satake isomorphism in characteristic p and establishes its equivalence to Herzig's Satake isomorphism for simply connected derived subgroups.

Findings

Constructed an isomorphism from dominant cocharacters to the Hecke algebra.

Proved the isomorphism is the inverse of Herzig's Satake isomorphism in specific cases.

Enhanced understanding of the structure of Hecke algebras in characteristic p.

Abstract

Let F be a local field with finite residue field of characteristic p and k an algebraic closure of the residue field. Let G be the group of F-points of a F-split connected reductive group. In the apartment corresponding to a chosen maximal split torus of T, we fix a hyperspecial vertex and denote by K the corresponding maximal compact subgroup of G. Given an irreducible smooth k-representation $\rho$ of K, we construct an isomorphism from the affine semigroup k-algebra of the dominant cocharacters of T onto the Hecke algebra $H(G, \rho)$. In the case when the derived subgroup of G is simply connected, we prove furthermore that our isomorphism is the inverse to the Satake isomorphism constructed by Herzig.