SQUARE-INTEGRABILITY OF INDUCED REPRESENTATIONS OF SEMIDIRECT PRODUCTS

We consider a semidirect product G = A x' H, with A abelian, and its u nitary representations of the form Ind(G0)(G) (x(0)m) where to is in t he dual group of A, G(0) is the stability group of x(0) and m is an ir reducible unitary representation of G(0) boolean AND H. We give a new selfcontained proof of the following result: the induced representatio n Ind(G0)(G) (x(0)m) is square-integrable if and only if the orbit G[x (0)] has nonzero Haar measure and m is square-integrable. Moreover we give an explicit form for the formal degree of IndS(G0)(G) (x(0)m).