Irreductibilité de certaines representations non unitaires dans des espaces de distributions propres

Abstract: Let ${\scriptstyle G_0 = K \times V}$ a semi-direct product of a compact Lie group ${\scriptstyle K}$ and a vector space ${\scriptstyle V}$. Let ${\scriptstyle {\cal E}_\chi}$ be the joint eigenspace (of functions or distributions) of the ${\scriptstyle G_0}$-invariant differential operators on ${\scriptstyle G_0/K}$ associated to a character ${\scriptstyle \chi}$ of the algebra of these operators. We give a sufficient condition for topological irreducibility of the action of ${\scriptstyle G_0}$ on ${\scriptstyle {\cal E}_\chi}$. This is a partial generalisation of a result of S. Helgason when ${\scriptstyle G_0}$ is a Cartan motion group. The present proof is of a more algebraic nature.

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