Admissibility For Quasiregular Representations of Exponential Solvable Lie Groups

2012/12/28 by Vignon Oussa, Oussa, Vignon
Mathematics · #22E25 #FOS: Mathematics #Functional Analysis (math.FA) #Representation Theory (math.RT) #math.FA #math.RT #msc:22E25

paper · pdf · doi:10.48550/arxiv.1212.6548

arxiv created 2013/04/27 · arxiv updated 2013/04/30

Abstract

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra \mathfrakn of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra \mathfrakh. Furthermore, let us assume that the linear adjoint action of \mathfrakh on \mathfrakn is diagonalizable with non-purely imaginary eigenvalues. Let τ=Ind%_H^{N\rtimes H} 1. We obtain an explicit direct integral decomposition for τ, including a description of the spectrum as a sub-manifold of (\mathfrakn+\mathfrakh)^∗, a formula for the multiplicity function of the unitary irreducible representations occurring in the direct integral, and a precise intertwining operator. Finally, we completely settle the admissibility question of τ. In fact, we show that if G=N\rtimes H is unimodular, then τ is never admissible, and if G is nonunimodular, τ is admissible if and only if the intersection of H and the center of G is equal to the identity of the group.

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