SKEW PRODUCT ACTIONS OF SEMI-DIRECT PRODUCT GROUPS

Abstract

Introduction Let f be a homomorphism from a second countable locally compact group G into the continuous automorphisms of a second countable compact group K. Let g ∗ k = f(g)(k) and suppose further the mapping given by (g, k) → g ∗ k is continuous from G × K into K, g ∈ G, k ∈ K. Then the cartesian product K × G becomes a group with multiplication given by (k1, g1)(k2, g2) = (k1(g1 ∗ k2), g1g2), where (ki, gi) ∈ K × G, i = 1, 2. We denote this semi-direct product group by K ×s G. Let (S, μ), a standard Borel space, be an ergodic Borel K ×s G-space with a probability invariant measure μ. In this paper we state necessary and sufficient conditions so the quotient Borel K ×s G-mapping from S into the space of K-orbits in S has relative discrete spectrum. These conditions are in terms of the stabilizers and the natural action of G on the dual K of K. (We remind the reader K is the set of unitary equivalence classes of irreducible unitary representations of K). Choose a Borel subset S of S that meets each K-orbit in S exactly once. Let p : (S, μ) → (S, μ) be a mapping given by p(s) = s where s is the unique point in S that meets the K-orbit s · K of s and μ = μ ◦ p−1.