### Abstract

The authors give a new method for calculating the spectrum and multiplicities of the irreducible unitary representations appearing in the quasi-regular representation U: N × L^{2}(ΓβN) → L^{2}(ΓβN) on a compact nilmanifold ΓβN. They proceed by decomposing the trace of U into traces of irreducible representations. The basic calculations in the paper deal with lattice subgroups (Λ = log Γ an additive lattice in the Lie algebra N), essentially using the Poisson summation formula. Let Ad′ be the contragredient adjoint action of N on N^{*}. If f{hook}_{0} ε{lunate} N^{*}, the multiplicity of π(f{hook}_{0}) in U is zero unless the Ad′(N) orbit of f{hook}_{0} meets Λ^{⊥} = {h ε{lunate} N^{*}: <h, Λ> ⊆ Z}. If f{hook}_{0} ε{lunate} Λ^{⊥}, then the multiplicity is a sum over representatives of certain Ad′(Γ)-orbits in, m(π(f{hook}_{0}),U) = ∑ Ad′(N)f{hook}_{0}∩Λ^{⊥} Ad′(Γ)k(f{hook}). The constants k(f{hook}) are given both algebraic and geometric interpretations that lead to simple and effective calculations. Similar formulas hold if Γ is not a lattice subgroup.

Original language | English (US) |
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Pages (from-to) | 123-154 |

Number of pages | 32 |

Journal | Journal of Functional Analysis |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1976 |

### ASJC Scopus subject areas

- Analysis

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## Cite this

*Journal of Functional Analysis*,

*21*(2), 123-154. https://doi.org/10.1016/0022-1236(76)90074-4