Solvability, Structure and Analysis for Minimal Parabolic Subgroups

TL;DR

This paper investigates the structure of minimal parabolic subgroups in real reductive Lie groups, focusing on cases where the Levi component is solvable or metabelian, and establishes foundational results for harmonic analysis and representation theory.

Contribution

It characterizes the structure of the Levi component in minimal parabolic subgroups, especially when solvable or metabelian, and derives explicit Plancherel and Fourier inversion formulas.

Findings

$rak{p}$ is solvable iff $M$ is commutative in linear groups.

$M$ is abelian modulo the center $Z_G$ in general cases.

Provides explicit Plancherel and Fourier inversion formulas.

Abstract

We examine the structure of the Levi component $MA$ in a minimal parabolic subgroup $P = MAN$ of a real reductive Lie group $G$ and work out the cases where $M$ is metabelian, equivalently where $\mathfrak{p}$ is solvable. When $G$ is a linear group we verify that $\mathfrak{p}$ is solvable if and only if $M$ is commutative. In the general case $M$ is abelian modulo the center $Z_G$, we indicate the exact structure of $M$ and $P$, and we work out the precise Plancherel Theorem and Fourier Inversion Formulae. This lays the groundwork for comparing tempered representations of $G$ with those induced from generic representations of $P$.