On the acceptable elements

TL;DR

This paper investigates the structure of acceptable elements in p-adic groups, proving the existence of a unique maximal element represented within the admissible subset of the Iwahori-Weyl group.

Contribution

It establishes the existence and uniqueness of the maximal acceptable element and characterizes it within the admissible subset of the Iwahori-Weyl group.

Findings

Unique maximal element exists in B(G, {μ})

Maximal element is represented in the admissible subset

Provides structural insight into acceptable elements

Abstract

In this paper, we study the set $B(G, \{\mu\})$ of acceptable elements for any $p$-adic group $G$. We show that $B(G, \{\mu\})$ contains a unique maximal element and the maximal element is represented by an element in the admissible subset of the associated Iwahori-Weyl group.