# Quasi-regular Representations of Two-Step Nilmanifolds

## Abstract

Let *G* be a connected and simply connected two-step nilpotent Lie group. If *H* is a closed subgroup of *G* such that *G* / *H* has an invariant measure, the operators of left translation by elements of *G* are unitary in \(L^2(G/H)\), giving rise to a unitary representation of *G*, called the quasiregular representation of *G* / *H*, and denoted by \({\text {Ind}}_H^G 1\). We say that two cocompact, discrete subgroups \(\Gamma _1\) and \(\Gamma _2\) of *G* are *representation equivalent* if the quasi-regular representations \(R_{\Gamma _1}\) and \(R_{\Gamma _2}\) are unitarily equivalent. We give in this paper an explicit intertwining operator between the two representations \(R_{\Gamma _1}\) and \(R_{\Gamma _2}\) induced from two representation equivalent subgroups \(\Gamma _1\) and \(\Gamma _2\).

### Keywords

Nilpotent lie group Uniform subgroup Unitary representation Polarization Disintegration Orbit Kirillov theory Intertwining operator### 1991 Mathematics Subject Classification:

22E25 22E40 22E27## Notes

### Acknowledgements

It is great pleasure to thank Professeur Jean Ludwig for beneficial conversations about the subject of the paper and many helpful suggestions and the anonymous referee for comments and suggestions.

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