  Using the calculus of Fourier integral operators on Lie groupoids developped
in [18], we study the fundamental solution of the evolution equation
($\partial$ $\partial$t + iP)u = 0 where P is a self adjoint elliptic order one
G-pseudodifferential operator on the Lie groupoid G. Along the way, we continue
the study of distributions on Lie groupoids done in [17] by adding the reduced
C *-algebra of G in the picture and we investigate the local nature of the
regularizing operators of [32].
