Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift L-X on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure lambda(G,chi), whose density with respect to the left Haar measure lambda(G) is a nontrivial positive character chi of G. We show that if p not equal 2 and G is amenable, then every L-p(lambda(G,chi)) spectral multiplier of L-X extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are L-p(lambda(G,chi)) spectral multipliers of L-X.