Harmonic Analysis in One-Parameter Metabelian Nilmanifolds

TL;DR

This paper studies harmonic analysis on one-parameter metabelian nilmanifolds, focusing on decomposing the quasi-regular representation into irreducibles and providing explicit formulas and operators.

Contribution

It provides a detailed orbital description of the spectrum, multiplicity function, and explicit intertwining operators for these nilmanifolds, with a direct computation of multiplicities.

Findings

Decomposition of the quasi-regular representation into irreducibles.

Explicit intertwining operator construction.

Direct multiplicity formula computation.

Abstract

Let $G$ be a connected, simply connected one-parameter metabelian nilpotent Lie group, that means, the corresponding Lie algebra has a one-codimensional abelian subalgebra. In this article we show that $G$ contains a discrete cocompact subgroup. Given a discrete cocompact subgroup $\Gamma$ of $G$, we define the quasi-regular representation $\tau = {\rm ind}_\Gamma^G 1$ of $G$. The basic problem considered in this paper concerns the decomposition of $\tau$ into irreducibles. We give an orbital description of the spectrum, the multiplicity function and we construct an explicit intertwining operator between $\tau$ and its desintegration without considering multiplicities. Finally, unlike the Moore inductive algorithm for multiplicities on nilmanifolds, we carry out here a direct computation to get the multiplicity formula.