Complex Fourier analysis on a nilpotent Lie group

Abstract

Let G G be a simply-connected nilpotent Lie group, with complexification G c {G_c} . The functions on G G which are analytic vectors for the left regular representation of G G on L 2 ( G ) {L_2}(G) are determined in this paper, via a dual characterization in terms of their analytic continuation to G c {G_c} , and by properties of their L 2 {L_2} Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case G = R G = R this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip.