Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

Abstract

This chapter discusses that induced representation always plays central and indispensable roles in the theory of group representations. This idea contributes greatly to the construction of irreducible representations. For example, every irreducible unitary representation of a simply connected nilpotent Lie group is obtained as an induced representation, and the Kirillov orbit method helps to construct it explicitly using polarization. Further, for semi-simple Lie groups, Langlands's classification of irreducible admissible representations relies largely upon the induction from parabolic subgroups. The chapter provides an overview on generalized Gelfand–Graev representations (GGGRs). These are induced representations, though far from being irreducible. GGGRs have been constructed for reductive algebraic groups over a finite field, through the Dynkin–Kostant theory on nilpotent classes. They form a series of representations induced from certain kinds of unipotent subgroups, parametrized by nilpotent orbits in the corresponding Lie algebras. The chapter also reviews that GGGRs are constructed analogously for reductive algebraic groups over an archimedian or non-archimedian local field.