Operators arising from representations of nilpotent Lie groups

The following two results are obtained for an irreducible multiplier representation T of a connected nilpotent Lie group. First, T_f is a Hilbert-Schmidt operator if f is square-integrable with compact support. Second, T_f is of trace class if f has derivatives with sufficiently many moments. An application is made of the latter result to show that T_f can be of trace class even when f is not continuous.