Some harmonic analysis on commutative nilmanifolds

TL;DR

This paper analyzes harmonic analysis on a family of commutative nilmanifolds formed by two-step nilpotent Lie groups and their automorphism groups, providing explicit inversion formulas, decompositions, and spherical functions.

Contribution

It extends previous work by deriving explicit inversion formulas and decompositions for square integrable cases, including the Heisenberg group, and parametrizes spherical functions for these pairs.

Findings

Explicit inversion formulas for N with square integrable representations

Decomposition of L^2(N) under K ⋉ N action for all Gelfand pairs

Explicit expressions for spherical functions in certain cases

Abstract

In this work, 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 2-step nilpotent Lie group have square integrable representations. In this 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 of $K \ltimes N$ on $L^{2}(N)$. This result completes the analysis carried out by Wolf, where the inversion formula is obtained in the case that $N$ has not square integrable representation. 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.