Representations of the $p$-Adic $GSpin_4$ and $GSpin_6$ and the Adjoint L-Function

TL;DR

This paper proves a conjecture relating the structure of generic L-packets for certain p-adic groups to the properties of the adjoint L-function, providing explicit formulas in terms of local Langlands L-functions.

Contribution

It confirms a conjecture by Gross and Prasad for p-adic GSpin groups of ranks 2 and 3, and explicitly describes the adjoint L-function for each L-packet.

Findings

Verification of the Gross-Prasad conjecture for specific p-adic groups.

Explicit formulas for the adjoint L-function in terms of local Langlands L-functions.

Characterization of generic L-packets via the adjoint L-function.

Abstract

We prove a conjecture of B. Gross and D. Prasad about determination of generic $L$-packets in terms of the analytic properties of the adjoint $L$-function for $p$-adic general even spin groups of semi-simple ranks 2 and 3. We also explicitly write the adjoint $L$-function for each $L$-packet in terms of the local Langlands $L$-functions for the general linear groups.