On local coefficients for non-generic representations of some classical groups

TL;DR

This paper investigates non-generic representations of classical groups over nonarchimedean fields, focusing on Bessel models, their properties under induction, and defining local coefficients for these models, extending classical results.

Contribution

It introduces a new analysis of Bessel models for non-generic representations, including an analogue of Rodier's theorem and the definition of local coefficients in this context.

Findings

Properties of Bessel models under induction are characterized.

An analogue of Rodier's theorem for Bessel models is proved.

Local coefficients are defined for supercuspidal and induced representations.

Abstract

This paper is concerned with representations of split orthogonal and quasi-split unitary groups over a nonarchimedean local field which are not generic, but which support a unique model of a different kind, the generalized Bessel model. The properties of the Bessel models under induction are studied, and an analogue of Rodier's theorem concerning the induction of Whittaker models is proved for Bessel models which are minimal in a suitable sense. The holomorphicity in the induction parameter of the Bessel functional is established. Last, local coefficients are defined for each irreducible supercuspidal representation which carries a Bessel functional and also for a certain component of each representation parabolically induced from such a supercuspidal.