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Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields

Yugo Takanashi

TL;DR

This work derives a unified parity formula linking the Langlands parameter of a conjugate self-dual discrete series representation of $GL_n$ over a $p$-adic field and the parity of its Jacquet–Langlands transfer to an inner form. Utilizing a globalizing method inspired by Prasad–Ramakrishnan and Mok’s unitary-base-change framework, the authors express the local parity as a product of local contributions and relate it to the Weil–Deligne parameter via $c(W_E,W_F,\sigma)$. The main theorem states that for $G= A^{\times}$ with $A=M_m(D)$ and appropriate $D$, the parity satisfies $c(G,G',\pi)=(-1)^{(n-1)ms} c(W_E,W_F,\sigma)^{ms}$, with a complementary variant using a related division algebra. The approach blends central simple algebras, global automorphic methods, Aubert–Zelevinsky involution invariance, and Jacquet–Langlands transfer to produce a broad parity correspondence generalizing prior special cases. These results advance understanding of how conjugate self-duality interacts with inner forms and Langlands parameters, with potential implications for automorphic descent and endoscopic transfer in unitary groups.

Abstract

We prove a general formula that relates the parity of the Langlands parameter of a conjugate self-dual discrete series representation of $\mathrm{GL}_n$ to the parity of its Jacquet-Langlands image. It gives a generalization of a partial result by Mieda concerning the case of invariant $1/n$ and supercuspidal representations. It also gives a variation of the result on the self-dual case by Prasad and Ramakrishnan.

Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields

TL;DR

This work derives a unified parity formula linking the Langlands parameter of a conjugate self-dual discrete series representation of over a -adic field and the parity of its Jacquet–Langlands transfer to an inner form. Utilizing a globalizing method inspired by Prasad–Ramakrishnan and Mok’s unitary-base-change framework, the authors express the local parity as a product of local contributions and relate it to the Weil–Deligne parameter via . The main theorem states that for with and appropriate , the parity satisfies , with a complementary variant using a related division algebra. The approach blends central simple algebras, global automorphic methods, Aubert–Zelevinsky involution invariance, and Jacquet–Langlands transfer to produce a broad parity correspondence generalizing prior special cases. These results advance understanding of how conjugate self-duality interacts with inner forms and Langlands parameters, with potential implications for automorphic descent and endoscopic transfer in unitary groups.

Abstract

We prove a general formula that relates the parity of the Langlands parameter of a conjugate self-dual discrete series representation of to the parity of its Jacquet-Langlands image. It gives a generalization of a partial result by Mieda concerning the case of invariant and supercuspidal representations. It also gives a variation of the result on the self-dual case by Prasad and Ramakrishnan.
Paper Structure (17 sections, 30 theorems, 29 equations)

This paper contains 17 sections, 30 theorems, 29 equations.

Table of Contents

  1. A notion of conjugate self-duality
  2. Basic definitions
  3. Parity associated with a conjugating pair
  4. Central simple algebra settings
  5. A construction of conjugating pairs
  6. Notations in local and global settings
  7. Global results what we use
  8. A globalizing result
  9. Special cases of the global Jacquet-Langlands correspondence
  10. Local results what we use
  11. Triviality of parity at almost all places
  12. A special case of the main theorem
  13. Parity invariance under the Aubert-Zelevinsky involution
  14. An auxiliary lemma
  15. Proof of the main theorem
  16. ...and 2 more sections

Key Result

Theorem 1

Let $D$ be a central division algebra over $F$ of rank $d$, $G$ be ${\mathrm {GL}}_m(D)$, and $\pi$ be a discrete series representation of $G$. Let $\sigma$ denote its Langlands parameter. Then there exists a parity relation

Theorems & Definitions (88)

  • Theorem 1: Prasad
  • Theorem 2: Mie
  • Theorem 3: Main Theorem
  • Definition 1.1
  • Example 1.2: Inclusion of the Weil groups associated with quadratic extensions
  • Example 1.3: Component-wise conjugation
  • Example 1.4: Switching components
  • Example 1.5: Non-trivial construction over local fields
  • Definition 1.6
  • Remark 1.7
  • ...and 78 more