A Satake homomorphism for the $\bmod \, p$ derived Hecke algebra

TL;DR

This paper develops a Satake homomorphism for the derived spherical Hecke algebra of a p-adic group with mod p coefficients, connecting it to the torus algebra and analyzing its properties.

Contribution

It introduces a Satake homomorphism for the derived Hecke algebra with mod p coefficients and studies its properties and image in degree one.

Findings

Established a Satake homomorphism for the derived Hecke algebra

Analyzed the image of the homomorphism in degree 1

Proved transitivity with respect to Levi subgroup inclusion

Abstract

We explore the structure of the derived spherical Hecke algebra of a $p$-adic group, a graded associative algebra whose degree $0$ subalgebra is the classical spherical Hecke algebra. Working with $\mathbb Z/ p^a$ coefficients, we establish a Satake homomorphism relating this graded algebra to the corresponding graded algebra for the torus. We investigate the image of this homomorphism in degree $1$, as well as other properties, such as transitivity with respect to inclusion of Levi subgroups.