Solvability, Structure and Analysis for Minimal Parabolic Subgroups

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 $P$.

References in corpus (2)

• The Plancherel Formula for Minimal Parabolic Subgroups
• Plancherel Formulae associated to Filtrations of Nilpotent Lie Groups