## Abstract

Let N be a simply connected nilpotent Lie group and Γ a discrete uniform subgroup. The authors consider irreducible representations σ in the spectrum of the quasi-regular representation N × L^{2}(Γ/N) → L^{2}(Γ→) which are induced from normal maximal subordinate subgroups M ⊆ N. The primary projection P_{σ} and all irreducible projections P ≤ P_{σ} are given by convolutions involving right Γ-invariant distributions D on Γ→, Pf(Γn) = D * f(Γn) = <D, n · f>all f ε{lunate} C^{∞}(Γ/N), where n · f(ζ) = f(ζ · n). Extending earlier work of Auslander and Brezin, and L. Richardson, the authors give explicit character formulas for the distributions, interpreting them as sums of characters on the torus T^{κ} = (Γ ∩ M) · [M, M]{minus 45 degree rule}M. By examining these structural formulas, they obtain fairly sharp estimates on the order of the distributions: if σ is associated with an orbit O ⊆ n^{*} and if V ⊆ n^{*} is the largest subspace which saturates θ in the sense that f ε{lunate} O ⇒ f + V ⊆ O. As a corollary they obtain Richardson's criterion for a projection to map C^{0}(Γ→) into itself. The authors also resolve a conjecture of Brezin, proving a Zero-One law which says, among other things, that if the primary projection P_{σ} maps C^{r}(Γ→) into C^{0}(Γ→), so do all irreducible projections P ≤ P_{σ}. This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in R^{n} lie in the coset ring of Z^{n} (the finitely additive Boolean algebra generated by cosets of subgroups in Z^{n}). This lemma may be useful in other investigations of nilmanifolds.

Original language | English (US) |
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Pages (from-to) | 255-284 |

Number of pages | 30 |

Journal | Journal of Functional Analysis |

Volume | 23 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1976 |

## ASJC Scopus subject areas

- Analysis

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