Spectral multipliers for sublaplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift LX on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure λG,χ , whose density with respect to the left Haar measure λG is a nontrivial positive character χ of G. We show that if p ̸= 2 and G is amenable, then every Lp (λG,χ ) spectral multiplier of LX 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 neces- sary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are Lp (λG,χ ) spectral multipliers of LX .