On the irreducibility of nonunitary induced representations of certain semidirect products

Abstract:

Let G be a connected Lie group which is a semidirect product of a compact subgroup K and a normal solvable subgroup S. Let $\Lambda$ be a character of S, and let ${M_\Lambda }$ be the stabilizer of $\Lambda$ in K. Let $[H,{\Lambda _\mu }]$ be a finite-dimensional irreducible representation of the subgroup $S{M_\Lambda }$ on the complex vector space H. In this paper we consider the induced representations of G on various Banach spaces, and study their topological irreducibility. The basic method used consists in studying the irreducibility of the Lie algebra representations which arise on the linear subspaces of K-finite vectors. The latter question then can be reduced to the problem of determining when certain modules over certain commutative algebras are irreducible. The method discussed in this paper leads to two theorems giving sufficient conditions on the character $\Lambda$ that the induced representations be topologically irreducible. The question of infinitesimal equivalence of various induced representations is also discussed.