Direct Integral Decompositions and Multiplicities for Induced Representations of Nilpotent Lie Groups

Abstract

Let A" be a Lie subgroup of the connected, simply connected nilpotent Lie group G, and let f, g be the corresponding Lie algebras.Suppose that a is an irreducible unitary representation of K. We give an explicit direct integral decomposition of IndA^c;o into irreducibles.The description uses the Kirillov orbit picture, which gives a bijection between GA and the coadjoint orbits in g* (and similarly for K A, f*).Let P: f * - g* be the canonical projection, let ff"cf* be the orbit corresponding to a, and, for ir e G A, let 0" c g* be the corresponding orbit.The main result of the paper says essentially that <rr e G A appears in the direct integral iff P~1(0a) meets C"; the multiplicity of m is the number of Ad*(K)-orbits in 0V n P'^(0a).There is also a natural description of the measure class in the integral.