On the characters of exponential groups

Let G be an exponential solvable group, O an orbit of the coadjoint representation and T the corresponding irreducible unitary representation of G. A polynomial function P, such that P ¦ O is positive and semi-invariant, determines a positive, self-adjoint operator A on the space of T. Using the resulution of singularities by H. Hironaka, one shows, under suitable conditions on O, that the function t → Tr(A^tT()A^t)( ε C_c^∞(G), fix) admits a meromorphic analytic continuation, with poles on the real axis.