Generalized Local Coefficients

TL;DR

This paper introduces generalized local coefficients for certain group representations, extending Shahidi's local coefficients to non quasi-split groups like $GL_m(D)$, and explores their properties and relations.

Contribution

It defines generalized local coefficients under specific assumptions, proves their equivalence to Shahidi's coefficients in the quasi-split case, and extends their applicability to non quasi-split groups.

Findings

Generalized local coefficients match Shahidi's coefficients in the quasi-split case.

They satisfy properties like relation to Plancherel measures and multiplicativity.

Applicable to $(Y,)$-generic representations, generalizing the notion of genericity.

Abstract

In this paper we showed that under two assumptions we are able to define interesting functions that we call generalized local coefficients. We showed that in the quasi-split case generalized local coefficients are up to a positive constant the same as Shahidi's local coefficients. We provide a proof that the non quasi-split group $GL_m(D)$, for a central division algebra $D$ satisfies those assumptions. We also showed that generalized local coefficients satisfy nice properties, like the relation to Plancherel measures and multiplicativity inherited by that of intertwining operators. Generalized local coefficients are only defined for representations that are $(Y,\varphi)$-generic which is a generalization of generic representations in the quasi-split case. Here $Y$ denotes a nilpotent element in the Lie algebra of the group and $\varphi$ is a co-character related to $Y$.