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Local Langlands Conjecture for $p$-adic $GSpin_4,$ $GSpin_6,$ and their inner forms

Mahdi Asgari, Kwangho Choiy

Abstract

We establish the local Langlands conjecture for small rank general spin groups $GSpin_4$ and $GSpin_6$ as well as their inner forms. We construct appropriate $L$-packets and prove that these $L$-packets satisfy the properties expected of them to the extent that the corresponding local factors are available. We are also able to determine the exact sizes of the $L$-packets in many cases.

Local Langlands Conjecture for $p$-adic $GSpin_4,$ $GSpin_6,$ and their inner forms

Abstract

We establish the local Langlands conjecture for small rank general spin groups and as well as their inner forms. We construct appropriate -packets and prove that these -packets satisfy the properties expected of them to the extent that the corresponding local factors are available. We are also able to determine the exact sizes of the -packets in many cases.

Paper Structure

This paper contains 25 sections, 17 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. The Preliminaries
  3. Notations and Conventions
  4. Group Structures
  5. Split Groups
  6. Inner Forms
  7. The $F$-points
  8. $L$-groups
  9. Local Langlands Conjecture in a General Setting
  10. Review of Results on Restriction
  11. Results of Gelbart-Knapp, Tadić, and Hiraga-Saito
  12. A Theorem of Labesse
  13. $L$-packets for Inner Forms of $SL_n$
  14. Local Langlands Correspondence for GSpin4 and its inner forms
  15. Construction of $L$-packets of $\mathrm{GSpin}_4$ and Its Inner Forms
  16. ...and 10 more sections

Key Result

Proposition 2.1

As algebraic groups over $F$ we have the following isomorphisms:

Theorems & Definitions (49)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 4.1
  • Proposition 4.2
  • Theorem 5.1
  • proof
  • Remark 5.2
  • Remark 5.3
  • ...and 39 more