
Journal of Lie Theory 30 (2020), No. 3, 673690 Copyright Heldermann Verlag 2020 Some Harmonic Analysis on Commutative Nilmanifolds Andrea L. Gallo FaMAF  CIEM  CONICET, Universidad Nacional, Córdoba 5000, Argentina andregallo88@gmail.com Linda V. Saal FaMAF  CIEM  CONICET, Universidad Nacional, Córdoba 5000, Argentina saal@mate.uncor.edu [Abstractpdf] We consider a family of Gelfand pairs $(K \ltimes N, N)$ (in short $(K,N)$) where $N$ is a two step nilpotent Lie group, and $K$ is the group of orthogonal automorphisms of $N$. This family has a nice analytic property: almost all these 2step nilpotent Lie group have square integrable representations. In these cases, following MooreWolf's theory, we find an explicit expression for the inversion formula of $N$, and as a consequence, we decompose the regular action of $K \ltimes N$ on $L^{2}(N)$. This explicit expression for the Fourier inversion formula of $N$, specialized to a class of commutative nilmanifolds described by J.\,Lauret, sharpens the analysis of J.\,A.\,Wolf in Section 14.5 in {\it Harmonic Analysis on Commutative Spaces} [Mathematical Surveys and Monographs 142, American Mathematical Society, Providence (2007)], and in {\it On the analytic structure of commutative nilmanifolds} [J. Geometric Analysis 26 (2016) 10111022], concerning the regular action of $K \ltimes N$ on $L^2(N)$. When $N$ is the Heisenberg group, we obtain the decomposition of $L^{2}(N)$ under the action of $K \ltimes N$ for all $K$ such that $(K,N)$ is a Gelfand pair. Finally, we also give a parametrization for the generic spherical functions associated to the pair $(K,N)$, and we give an explicit expression for these functions in some cases. Keywords: Gelfand pairs, inversion formula, nilpotent Lie group, regular representation. MSC: 43A80, 22E25. [ Fulltextpdf (190 KB)] for subscribers only. 