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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 p is solvable. When G is a linear group we verify that p is solvable if and only if M is commutative. In the general case M is abelian modulo the center ZG , 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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