$R$--groups, elliptic representations, and parameters for $GSpin$ groups

TL;DR

This paper analyzes the structure of R-groups and elliptic representations for GSpin groups over p-adic fields, establishing their properties and classifications, including the triviality of cocycles and isomorphisms with Arthur R-groups.

Contribution

It provides a detailed description of R-groups for GSpin groups, proves the triviality of associated cocycles, and classifies elliptic tempered spectra, linking Knapp-Stein and Arthur R-groups.

Findings

Knapp-Stein R-groups are elementary two groups.

Cocycle associated with R-groups is trivial, ensuring multiplicity one.

Arthur R-groups for GSpin_{2n+1} are isomorphic to Knapp-Stein R-groups.

Abstract

We study parabolically induced representations for $GSpin_m(F)$ with $F$ a $p$--adic field of characteristic zero. The Knapp-Stein $R$--groups are described and shown to be elementary two groups. We show the associated cocycle is trivial proving multiplicity one for induced representations. We classify the elliptic tempered spectrum. For $GSpin_{2n+1}(F)$, we describe the Arthur (Endoscopic) $R$--group attached to Langlands parameters, and show these are isomorphic to the corresponding Knapp-Stein $R$--groups.