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

TL;DR

This paper establishes a general formula linking the parity of Langlands parameters of conjugate self-dual representations of inner forms of GL_n over p-adic fields to their Jacquet-Langlands images, extending previous partial results.

Contribution

It provides a comprehensive formula connecting the parity of Langlands parameters with Jacquet-Langlands images for conjugate self-dual representations, generalizing prior specific cases.

Findings

Derived a formula relating Langlands parameter parity to Jacquet-Langlands images.

Extended previous results to broader classes of representations.

Unified the understanding of parity phenomena in the context of inner forms of GL_n.

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.