A New Bound for the Uniform Admissibility Theorem

TL;DR

This paper revisits Bernstein's uniform bound on the dimension of K-fixed vectors in irreducible admissible representations of p-adic groups, providing a sharper bound through a new proof of a key lemma.

Contribution

It offers a new, more precise bound for the uniform admissibility constant by presenting an alternative proof of a crucial lemma in Bernstein's theorem.

Findings

Derived a sharper bound for N(G,K)

Revised proof of a key lemma in Bernstein's theorem

Enhanced understanding of admissibility bounds in p-adic groups

Abstract

In "All p-adic reductive groups are tame" Bernstein proved that for a reductive group G over a local non-archimedean field F and a compact open subgroup K of G there exists a uniform bound N(G,K) such that for every irreducible, smooth, and admissible representation V of G the dimension of the subspace of K-fixed vectors in V is bounded by N(G,K). In this note I repeat the proof of Bernstein and give my proof to one of the two main lemmas. The new proof of this lemma gives a new, sharper bound for the constant N(G,K).