Solvability,Structure, and Analysis for Minimal Parabolic Subgroups

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.