Unicity of types for supercuspidal representations of p-adic SL(2)

TL;DR

This paper investigates the uniqueness of types for supercuspidal representations of p-adic SL(2), introducing archetypes and establishing their relation to GL(2) types, with implications for the inertial Langlands correspondence.

Contribution

It introduces the notion of archetypes for SL(2) supercuspidal representations and shows they are induced from Bushnell--Kutzko types, linking archetypes between SL(2) and GL(2).

Findings

Any archetype in SL(2) is restricted from one in GL(2).

The number of archetypes for a supercuspidal representation is explicitly described in terms of ramification.

A relationship between archetypes for GL(2) and SL(2) is established via L-packets, leading to an inertial Langlands correspondence.

Abstract

We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of $\mathbf{SL}_2$ over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the $\mathbf{SL}_2$-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in $\mathbf{SL}_2$ is restricted from one in $\mathbf{GL}_2$. From this it follows that any archetype must be induced from a Bushnell--Kutzko type. Given a supercuspidal representation $\pi$, we give an additional explicit description of the number of archetypes admitted by $\pi$ in terms of its ramification. We also describe a relationship between archetypes for $\mathbf{GL}_2$ and $\mathbf{SL}_2$ in terms of $L$-packets, and deduce an inertial Langlands correspondence for $\mathbf{SL}_2$.