On $L$-packets and depth for $SL_2(K)$ and its inner form

TL;DR

This paper investigates the relationship between the depth of irreducible representations of SL_2(K) over a characteristic two local field and their Langlands parameters, providing classifications and explicit depth formulas.

Contribution

It establishes a depth inequality for representations and parameters, characterizes when equality holds, and classifies all L-packets for SL_2(K) and its inner form with explicit depth formulas.

Findings

Depth of irreducible representations exceeds that of L-parameters unless the parameter is essentially tame.

Complete classification of L-packets for SL_2(K) and its inner form.

Explicit formulas for the depths of L-parameters.

Abstract

We consider the group $SL_2(K)$, where $K$ is a local non-archimedean field of characteristic two. We prove that the depth of any irreducible representation of $SL_2 (K)$ is larger than the depth of the corresponding Langlands parameter, with equality if and only if the L-parameter is essentially tame. We also work out a classification of all $L$-packets for $SL_2 (K)$ and for its non-split inner form, and we provide explicit formulae for the depths of their $L$-parameters.