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

Anne-Marie Aubert, Roger Plymen

TL;DR

This work analyzes depth-zero, non-archimedean, local fields $F$ with $p>2$ to compare endoscopic character identities for $G= ext{SL}_2(F)$ against Sally–Shalika’s classical character formulas. It centers on the depth-zero supercuspidal $L$-packet of size $4$, showing how norm-$1$ groups $H_1,H_2,H_3$ in the three quadratic extensions of $F$ govern the endoscopic identities, including explicit transfer factors and little parameters. The authors treat both regular and non-regular Langlands parameters, deriving endoscopic expressions that match Sally–Shalika’s character values on tori and elliptic elements, and they demonstrate stability of the stable characters across inner forms of $ ext{SL}_2$. Overall, the paper confirms the compatibility of endoscopic character identities with the Sally–Shalika formulas for $ ext{SL}_2$ and clarifies the involvement of all three quadratic endoscopic groups in the size-$4$ packet. This advances understanding of endoscopy in low-rank cases and provides explicit computational checks for the transfer factors and stable distributions in this setting.

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 $\mathrm{SL}_2(F)$.

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

TL;DR

This work analyzes depth-zero, non-archimedean, local fields with to compare endoscopic character identities for against Sally–Shalika’s classical character formulas. It centers on the depth-zero supercuspidal -packet of size , showing how norm- groups in the three quadratic extensions of govern the endoscopic identities, including explicit transfer factors and little parameters. The authors treat both regular and non-regular Langlands parameters, deriving endoscopic expressions that match Sally–Shalika’s character values on tori and elliptic elements, and they demonstrate stability of the stable characters across inner forms of . Overall, the paper confirms the compatibility of endoscopic character identities with the Sally–Shalika formulas for and clarifies the involvement of all three quadratic endoscopic groups in the size- packet. This advances understanding of endoscopy in low-rank cases and provides explicit computational checks for the transfer factors and stable distributions in this setting.

Abstract

We consider the depth-zero supercuspidal -packets of where is a non-archimedean local field of characteristic zero. We compare the explicit endoscopic character identities for with the classical character formulas of Sally-Shalika. Our main result concerns the supercuspidal -packet of size . For this -packet, we show how the norm groups in the three quadratic extensions of play a crucial role in the endoscopic character identities for .

Paper Structure

This paper contains 13 sections, 14 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Elliptic tori
  3. Elliptic tori
  4. Weyl discriminant
  5. The function $f$
  6. Elliptic torus in $\mathrm{SL}_2(\mathfrak{f})$
  7. Depth-zero supercuspidal representations of $\mathrm{SL}_2(F)$
  8. Virtual characters far from the identity: the non-regular parameter
  9. Virtual characters near the identity: the non-regular parameter
  10. Virtual characters: regular parameters
  11. Endoscopy: the non-regular parameter
  12. Endoscopy: regular parameters
  13. Stability across the inner forms of $\mathrm{SL}_2$

Key Result

Theorem 1.8

Let $\gamma$ be a regular, semisimple element in $T^\varepsilon$. Let $\Pi_\phi(\mathrm{SL}_2(F))$ be the depth-zero supercuspidal $L$-packet with regular $L$-parameter $\phi$. Let $s$ in (endo) denote the non-trivial element in $S_\phi$. Then the endoscopic group $H$ is $E^1$. The little parameter and

Theorems & Definitions (26)

  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.10
  • proof
  • Theorem 4.3
  • proof
  • Theorem 4.10
  • proof
  • Theorem 5.1
  • proof
  • ...and 16 more