CHARACTERISTICS OF SOLVABLE LIE-GROUPS

We consider a connected, solvable, unimodular Lie group G. Let g be the Lie algebra of G. Let l be in the dual of g. Under the assumption that g(l) is reductive in g, we construct an application phi --> F(l),phi from D(G) to the space of C infinity functions on an open dense subset of G(l). Using this application we give a formula for the trace of the operator n(l, G)(phi), where n(l, G) is the unitary representation of G associated to l. This formula applies to the square-integrable representations modulo Z(G) of the group G.