Rational subspaces of induced representations and nilmanifolds

Abstract

Recently, Auslander and Brezin developed a technique of distinguishing between certain unitarily equivalent irreducible subspaces of L 2 {L^2} of the Heisenberg nilmanifold. In this paper we extend the Auslander-Brezin technique to arbitrary induced representations of arbitrary locally compact groups. We then return to nilmanifolds, showing that the existence of a “nice” theory of distinguished subspaces is equivalent to the existence of square integrable representations for the group.