Bounds on multiplicities of spherical spaces over finite fields -- the general case

TL;DR

This paper extends bounds on multiplicities of spherical spaces over finite fields from type A groups to general connected reductive groups, confirming a conjecture and broadening the scope of previous results.

Contribution

It generalizes existing bounds on multiplicities from type A groups to all connected reductive groups over integers, proving Conjecture A.

Findings

Bounded multiplicities for spherical spaces over finite fields.

Generalization from type A to all connected reductive groups.

Proof of Conjecture A in the broader setting.

Abstract

Let $G$ be a connected reductive group scheme acting on a spherical scheme $X$. In the case where $G$ is of type $A_n$, Aizenbud and Avni proved the existence of a number $C$ such that the multiplicity $\dim\hom(\rho,\mathbb{C}[X(F)])$ is bounded by $C$, for any finite field $F$ and any irreducible representation $\rho$ of $G(F)$. In this paper, we generalize this result to the case where $G$ is a connected reductive group scheme over $\mathbb{Z}$, and prove Conjecture A of [1].