Ultra-irreducibility of induced representations of semidirect products

Abstract:

Let the Lie group $G$ be a semidirect product, $G = SK$, of a connected, closed, normal subgroup $S$ and a closed subgroup $K$. Let $\Lambda$ be a nonunitary character of $S$, and let ${K_\Lambda }$ be its stability subgroup in $K$. Let ${I^{\Lambda \mu }}$, for any irreducible representation $\mu$ of ${K_\Lambda }$, denote the representation ${I^{\Lambda \mu }}$ of $G$ induced by the representation $\Lambda \mu$ of $S{K_\Lambda }$. The representation spaces are subspaces of the distributions. We show that ${I^{\Lambda \mu }}$ is ultra-irreducible when the corresponding Poisson transform is injective, and find a sufficient condition for this injectivity.