Comparing Bushnell-Kutzko and S\'echerre's constructions of types for $\mathrm{GL}_{N}$ and its inner forms with Yu's construction

TL;DR

This paper compares two different constructions of types for supercuspidal representations of general linear groups and their inner forms over non-archimedean local fields, establishing their equivalence in the essentially tame case.

Contribution

It demonstrates the equivalence between Bushnell-Henniart and Yu's constructions for essentially tame supercuspidal representations of AG, connecting two major approaches.

Findings

Bushnell-Henniart supercuspidal representations are equivalent to Yu's tame supercuspidal representations.

The paper clarifies the relationship between simple types and Yu's construction.

It bridges the gap between different methods of constructing supercuspidal representations.

Abstract

Let $F$ be a non-archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. To construct types for supercuspidal representations of $G$, simple types by S\'echerre and Yu's construction are already known. In this paper, we compare these constructions. In particular, we show essentially tame supercuspidal representations of $G$ defined by Bushnell-Henniart are nothing but tame supercuspidal representations defined by Yu.