Bounds on multiplicities of spherical spaces over finite fields

TL;DR

This paper establishes a uniform bound on the multiplicities of irreducible representations in the space of functions on spherical schemes over finite fields, providing explicit bounds and conjecturing broader applicability.

Contribution

It proves a boundedness result for multiplicities in spherical spaces over finite fields and offers explicit bounds, extending understanding of representation multiplicities.

Findings

Existence of a uniform bound C for multiplicities.

Explicit calculation of the bound C.

Conjecture extending results to all reductive groups and local fields.

Abstract

Let $G$ be a reductive group scheme of type $A$ acting on a spherical scheme $X$. We prove that there exists 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 $G(F)$. We give an explicit bound for $C$. We conjecture that this result is true for any reductive group scheme and when $F$ ranges (in addition) over all local fields of characteristic $0$.