Solvability, Structure and Analysis for Minimal Parabolic Subgroups

2016/10/25 by Joseph A. Wolf, Wolf, Joseph A.
Mathematics · #22E27 #22E45 #43A80 #FOS: Mathematics #Functional Analysis (math.FA) #Representation Theory (math.RT) #math.FA #math.RT #msc:22E27 #msc:22E45 #msc:43A80

paper · pdf · doi:10.48550/arxiv.1610.08105

arxiv created 2017/01/22 · arxiv updated 2017/01/24

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 \mathfrakp is solvable. When G is a linear group we verify that \mathfrakp 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.

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