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

L. Corwin, F. P. Greenleaf

Research output: Contribution to journalArticle

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 × 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.

Original languageEnglish (US)
Pages (from-to)255-284
Number of pages30
JournalJournal of Functional Analysis
Volume23
Issue number3
DOIs
StatePublished - 1976

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Nilmanifolds
Integral Formula
Projection
kernel
Coset
Subgroup
Lemma
Zero-one Law
Graph Polynomial
Character Formula
Integral Points
Invariant Distribution
Nilpotent Lie Group
Boolean algebra
Irreducible Representation
Thing
Explicit Formula
Convolution
Resolve
Torus

ASJC Scopus subject areas

  • Analysis

Cite this

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

In: Journal of Functional Analysis, Vol. 23, No. 3, 1976, p. 255-284.

Research output: Contribution to journalArticle

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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.

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