Some harmonic analysis on commutative nilmanifolds

Abstract

In this work, we consider a family of Gelfand pairs (KnN, N) (inshort (K, N) ) where Nis a two step nilpotent Lie group, and Kis the group oforthogonal automorphisms ofN. This family has a nice analytic property: almos tall these 2-step nilpotent Lie group have square integrable representations. In these cases, following Moore-Wolf’s theory, we find an explicit expression for the inversion formula of N, and as a consequence, we decompose the regular action ofKnNonL2(N). This explicit expression for the Fourier inversion formula of N, specializedto a class of commutative nilmanifolds described by J. Lauret, sharpens the recent analysis due to J. Wolf concerning the regular action ofKnNonL2(N) . When Nis the Heisenberg group, we obtain the decomposition ofL2(N) under the action of KnN for all Ksuch that (K, N) is a Gelfand pair. Finally, we also give aparametrization for the generic spherical functions associated to the pair (K, N) ,and we give an explicit expression for these functions in some cases