On the composition series of the standard Whittaker (g,K)-modules

TL;DR

This paper investigates the structure and composition series of standard Whittaker (g,K)-modules for real reductive Lie groups, providing explicit descriptions in generic and specific integral cases, notably for U(n,1).

Contribution

It determines the structure of standard Whittaker (g,K)-modules for generic infinitesimal characters and explicitly describes their composition series for U(n,1) with regular integral characters.

Findings

Structures of standard Whittaker (g,K)-modules are characterized for generic infinitesimal characters.

Explicit composition series are derived for U(n,1) with regular integral infinitesimal characters.

Provides a detailed analysis of Whittaker modules in both generic and integral cases.

Abstract

For a real reductive linear Lie group G, the space of Whittaker functions is the representation space induced from a non-degenerate unitary character of the Iwasawa nilpotent subgroup. Defined are the standard Whittaker (g,K)-modules, which are K-admissible submodules of the space of Whittaker functions. We first determine the structures of them when the infinitesimal characters characterizing them are generic. As an example of the integral case, we determine the composition series of the standard Whittaker (g,K)-module when G is the group U(n,1) and the infinitesimal character is regular integral.