Sharpness versus bluntness in affine Weyl groups

TL;DR

This paper explores the relationship between geometric and arithmetic diagrams in the context of supercuspidal unipotent representations of simple p-adic groups, providing an algebraic explanation for their bijection.

Contribution

It offers a purely algebraic explanation for the bijection between geometric and arithmetic diagrams associated with certain p-adic group representations.

Findings

Provides an algebraic framework for understanding the bijection

Includes additional material in the second version

Enhances understanding of supercuspidal unipotent representations

Abstract

We give an explanation of the bijection between geometric and arithmetic diagrams attached to supercuspidal unipotent representations of a simple p-adic group which is based purely on algebra. The second version contains much additional material compared to the first version.