Comparison of the Sally-Shalika character formulas with the endoscopic character identities for $\mathrm{SL}_2$

TL;DR

This paper compares Sally-Shalika character formulas with endoscopic character identities for depth-zero supercuspidal $L$-packets of $ ext{SL}_2$ over non-archimedean fields, highlighting the role of norm 1 groups in the supercuspidal case.

Contribution

It explicitly relates Sally-Shalika formulas to endoscopic identities for a specific supercuspidal $L$-packet of size 4 in $ ext{SL}_2$, emphasizing the importance of norm 1 groups in quadratic extensions.

Findings

Endoscopic character identities are explicitly compared with Sally-Shalika formulas.

The supercuspidal $L$-packet of size 4 is analyzed in detail.

Norm 1 groups in quadratic extensions are crucial in the identities.

Abstract

We consider the depth-zero supercuspidal $L$-packets of $\mathrm{SL}_2(F)$ where $F$ is a non-archimedean local field of characteristic zero. We compare the explicit endoscopic character identities for $\mathrm{SL}_2(F)$ with the classical character formulas of Sally-Shalika. Our main result concerns the supercuspidal $L$-packet of size $4$. For this $L$-packet, we show how the norm $1$ groups $H_1, H_2, H_3$ in the three quadratic extensions of $F$ play a crucial role in the endoscopic character identities for $\mathrm{SL}_2(F)$.