### 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) |
---|---|

Pages (from-to) | 255-284 |

Number of pages | 30 |

Journal | Journal of Functional Analysis |

Volume | 23 |

Issue number | 3 |

DOIs | |

State | Published - 1976 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*23*(3), 255-284. https://doi.org/10.1016/0022-1236(76)90051-3

**Integral formulas with distribution kernels for irreducible projections in L2 of a nilmanifold.** / Corwin, L.; Greenleaf, F. P.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 23, no. 3, pp. 255-284. https://doi.org/10.1016/0022-1236(76)90051-3

}

TY - JOUR

T1 - Integral formulas with distribution kernels for irreducible projections in L2 of a nilmanifold

AU - Corwin, L.

AU - Greenleaf, F. P.

PY - 1976

Y1 - 1976

N2 - 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 × L2(Γ/N) → L2(Γ→) which are induced from normal maximal subordinate subgroups M ⊆ N. The primary projection Pσ and all irreducible projections P ≤ Pσ 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 Tκ = (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 C0(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 Pσ maps Cr(Γ→) into C0(Γ→), so do all irreducible projections P ≤ Pσ. This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in Rn lie in the coset ring of Zn (the finitely additive Boolean algebra generated by cosets of subgroups in Zn). This lemma may be useful in other investigations of nilmanifolds.

AB - 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 × L2(Γ/N) → L2(Γ→) which are induced from normal maximal subordinate subgroups M ⊆ N. The primary projection Pσ and all irreducible projections P ≤ Pσ 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 Tκ = (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 C0(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 Pσ maps Cr(Γ→) into C0(Γ→), so do all irreducible projections P ≤ Pσ. This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in Rn lie in the coset ring of Zn (the finitely additive Boolean algebra generated by cosets of subgroups in Zn). This lemma may be useful in other investigations of nilmanifolds.

UR - http://www.scopus.com/inward/record.url?scp=49549129003&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49549129003&partnerID=8YFLogxK

U2 - 10.1016/0022-1236(76)90051-3

DO - 10.1016/0022-1236(76)90051-3

M3 - Article

AN - SCOPUS:49549129003

VL - 23

SP - 255

EP - 284

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 3

ER -