Socle filtrations of the standard Whittaker (g,K)-modules of Spin(r,1)

TL;DR

This paper investigates the detailed internal structure of standard Whittaker modules for the group Spin(r,1) with regular integral infinitesimal character, expanding understanding beyond the generic case.

Contribution

It explicitly determines the socle filtrations of these modules in the integral case, a previously less understood scenario in representation theory.

Findings

Complete socle filtrations for Spin(r,1) modules with integral infinitesimal character

Clarification of module structures in the integral case

Extension of known results from generic to integral infinitesimal characters

Abstract

Studied are the composition series of the standard Whittaker (g,K)-modules. For a generic infinitesimal character, the structures of these modules are completely understood, but if the infinitesimal character is integral, then there are not so many cases in which the structures of them are known. In this paper, as an example of the integral case, we determine the socle filtrations of the standard Whittaker (g,K)-modules when G is the group Spin(r,1) and the infinitesimal character is regular integral.