Supercuspidal characters of $\operatorname{SL}_2$ over a $p$-adic field

TL;DR

This paper computes explicit supercuspidal character formulas for SL_2 over p-adic fields using modern techniques, including the Moy-Prasad theory, providing detailed character tables and novel insights into exceptional cases.

Contribution

It offers the first explicit calculations of supercuspidal characters for SL_2 over p-adic fields, especially for exceptional depth-zero cases, advancing p-adic harmonic analysis.

Findings

Explicit character tables for supercuspidal representations

Computation of exceptional supercuspidal characters

Application of Moy-Prasad theory to p-adic harmonic analysis

Abstract

The character formulas of Sally and Shalika are an early triumph in $p$-adic harmonic analysis, but, to date, the calculations underlying the formulas have not been available. In this paper, which should be viewed as a precursor of the forthcoming volume by the authors and Alan Roche, we leverage modern technology (for example, the Moy-Prasad theory) to compute explicit character tables. An interesting highlight is the computation of the 'exceptional' supercuspidal characters, i.e., those depth-zero representations not arising by inflation-induction from a Deligne-Lusztig representation of finite $\operatorname{SL}_2$; this provides a concrete application for the recent work of DeBacker and Kazhdan.