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Twisted endoscopic character relation for toral supercuspidal L-packets of classical groups

Masao Oi

Abstract

We prove that Kaletha's toral supercuspidal L-packets satisfy the twisted endoscopic character relation in some cases, including the case of general linear groups equipped with an involution. Consequently, we verify that Kaletha's construction of the local Langlands correspondence for toral supercuspidal representations of quasi-split symplectic or special orthogonal groups coincides with Arthur's. The strategy is to emulate Kaletha's proof of the standard endoscopic character relation in the twisted setting by appealing to Waldspurger's framework ``l'endoscopie tordue n'est pas si tordue''.

Twisted endoscopic character relation for toral supercuspidal L-packets of classical groups

Abstract

We prove that Kaletha's toral supercuspidal L-packets satisfy the twisted endoscopic character relation in some cases, including the case of general linear groups equipped with an involution. Consequently, we verify that Kaletha's construction of the local Langlands correspondence for toral supercuspidal representations of quasi-split symplectic or special orthogonal groups coincides with Arthur's. The strategy is to emulate Kaletha's proof of the standard endoscopic character relation in the twisted setting by appealing to Waldspurger's framework ``l'endoscopie tordue n'est pas si tordue''.
Paper Structure (72 sections, 85 theorems, 312 equations)

This paper contains 72 sections, 85 theorems, 312 equations.

Table of Contents

  1. Introduction
  2. Notation and assumptions on $p$
  3. Notation
  4. $p$-adic fields
  5. Algebraic varieties and algebraic groups
  6. Centralizers and normalizers
  7. Reductive groups
  8. Bruhat--Tits theory
  9. Finite sets with Galois actions
  10. Several arithmetic invariants
  11. Finite fields
  12. Assumptions on $p$
  13. Kaletha's regular supercuspidal local Langlands correspondence
  14. Regular supercuspidal representations
  15. Several invariants for the construction of LLC
  16. ...and 57 more sections

Key Result

Theorem 1.1

Let $\mathbf{H}$ be a quasi-split special orthogonal or symplectic group over $F$. Suppose that $p$ is sufficiently large. The Local Langlands correspondences of Arthur and Kaletha coincide for any "toral" supercuspidal representation of $H\colonequals \mathbf{H}(F)$.

Theorems & Definitions (188)

  • Theorem 1.1: Theorem \ref{['thm:Arthur=Kaletha']}
  • Theorem 1.3: Theorems \ref{['thm:spec-well-def-GL']}, \ref{['thm:spec-well-def-unram']}
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4: Kal19
  • Remark 3.5
  • Definition 3.6
  • Remark 3.7
  • Remark 3.8
  • ...and 178 more