[Paper Review] On local coefficients for non-generic representations of some classical groups

Solomon Friedberg, David Theo Goldberg|arXiv (Cornell University)|Feb 13, 1996

Advanced Algebra and Geometry8 references3 citations

TL;DR

This paper studies non-generic representations of split orthogonal and quasi-split unitary groups over nonarchimedean local fields that support a unique generalized Bessel model instead of a Whittaker model. It establishes the holomorphicity of Bessel functionals under induction, proves an analogue of Rodier's theorem for minimal Bessel models, and defines local coefficients for supercuspidal representations with Bessel functionals and their induced components, advancing the theory of non-generic automorphic forms in classical groups.

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.

Motivation & Objective

• To extend the theory of local coefficients beyond generic representations to non-generic ones in classical groups.
• To study the behavior of generalized Bessel models under parabolic induction.
• To establish the holomorphicity of Bessel functionals in the induction parameter.
• To define local coefficients for irreducible supercuspidal representations admitting a Bessel functional.
• To generalize Rodier's theorem on Whittaker models to the setting of minimal Bessel models.

Proposed method

• Analyzes the structure of irreducible supercuspidal representations that support a unique generalized Bessel model instead of a Whittaker model.
• Applies parabolic induction to construct representations with Bessel functionals from supercuspidal ones.
• Proves that the Bessel functional is holomorphic in the induction parameter using analytic techniques in representation theory.
• Establishes a criterion for minimality of Bessel models analogous to Rodier's criterion for Whittaker models.
• Defines local coefficients via the functional equations of Bessel functionals in the induced setting.
• Uses the uniqueness of the Bessel model to ensure well-definedness of the coefficient system.

Experimental results

Research questions

• RQ1 How do generalized Bessel models behave under parabolic induction in classical groups?
• RQ2 What conditions ensure the holomorphicity of Bessel functionals in the induction parameter?
• RQ3 Can an analogue of Rodier's theorem for Whittaker models be established for minimal Bessel models?
• RQ4 How can local coefficients be defined for non-generic representations with Bessel functionals?
• RQ5 What is the structure of the component of a parabolically induced representation that carries a Bessel functional?

Key findings

• The Bessel functional associated with a parabolically induced representation is holomorphic in the induction parameter.
• An analogue of Rodier's theorem holds for minimal Bessel models, characterizing irreducibility via the uniqueness of the model.
• Local coefficients are well-defined for each irreducible supercuspidal representation admitting a Bessel functional.
• Local coefficients are also defined for a specific component of each representation parabolically induced from such a supercuspidal representation.
• The generalized Bessel model provides a unique model for non-generic representations, replacing the role of the Whittaker model.
• The theory of local coefficients is extended beyond generic representations to include non-generic ones via the Bessel model framework.