Constant term functors with $\mathbb{F}_p$-coefficients

TL;DR

This paper investigates the constant term functor for $F_p$-sheaves on the affine Grassmannian, revealing its tensor structure and implications for mod $p$ Satake theory, with geometric proofs of key results.

Contribution

It establishes that the constant term functor induces a tensor functor between categories of equivariant perverse $F_p$-sheaves, providing new insights into the mod $p$ Satake transform.

Findings

The constant term functor is a tensor functor between categories of equivariant perverse $F_p$-sheaves.

Derived geometric proofs for results on the mod $p$ Satake transform.

Gained understanding of the structure of the space of mod $p$ Satake parameters.

Abstract

We study the constant term functor for $\mathbb{F}_p$-sheaves on the affine Grassmannian in characteristic $p$ with respect to a Levi subgroup. Our main result is that the constant term functor induces a tensor functor between categories of equivariant perverse $\mathbb{F}_p$-sheaves. We apply this fact to get information about the Tannakian monoids of the corresponding categories of perverse sheaves. As a byproduct we also obtain geometric proofs of several results due to Herzig on the mod $p$ Satake transform and the structure of the space of mod $p$ Satake parameters.