On the Moore Formula of Compact Nilmanifolds

TL;DR

This paper presents a new multiplicity formula for irreducible unitary representations in the decomposition of quasi-regular representations of two-step nilpotent Lie groups, extending Moore's formula from the Abelian case.

Contribution

It introduces a novel multiplicity formula for the decomposition of quasi-regular representations of two-step nilpotent Lie groups, generalizing Moore's formula beyond Abelian groups.

Findings

New multiplicity formula for two-step nilpotent Lie groups

Extension of Moore's formula from Abelian to non-Abelian cases

Enhanced understanding of representation decomposition in nilmanifolds

Abstract

Let $G$ be a connected and simply connected two-step nilpotent Lie group and $\Gamma$ a lattice subgroup of $G$. In this note, we give a new multiplicity formula, according to the sense of Moore, of irreducible unitary representations involved in the decomposition of the quasi-regular representation ${\rm Ind}_{\Gamma}^G(1)$. Extending then the Abelian case.