The socle filtrations of principal series representations of $SL(3,\mathbb{R})$ and $Sp(2,\mathbb{R})$

TL;DR

This paper provides a detailed analysis of the socle filtrations of principal series representations for $SL(3,R)$ and $Sp(2,R)$, revealing their structure when induced from minimal parabolics with nonsingular infinitesimal character.

Contribution

It offers a complete description of the socle filtrations of these modules, expanding understanding beyond the known composition factors.

Findings

Explicit socle filtration structures determined

Connections to Kazhdan-Lusztig-Vogan conjecture confirmed

Enhanced understanding of $(rak{g},K)$-modules for these groups

Abstract

We study the structure of the $(\mathfrak{g},K)$-modules of the principal series representations of $SL(3,\mathbb{R})$ and $Sp(2,\mathbb{R})$ induced from minimal parabolic subgroups, in the case when the infinitesimal character is nonsingular. The composition factors of these modules are known by Kazhdan-Lusztig-Vogan conjecture. In this paper, we give complete descriptions of the socle filtrations of these modules.