L2-singular dichotomy for orbital measures of classical simple Lie algebras

Sanjiv Kumar Gupta; Kathryn E. Hare; Sobhan Seyfaddini

doi:10.1007/s00209-008-0364-z

L²-singular dichotomy for orbital measures of classical simple Lie algebras

Sanjiv Kumar Gupta, Kathryn E. Hare, Sobhan Seyfaddini

Mathematics

Research output: Contribution to journal › Article › peer-review

13 Citations (Scopus)

Abstract

We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μ^k is singular to Lebesgue measure or μ^k ∈ L². The minimum k for which μ^k ∈ L² is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.

Original language	English
Pages (from-to)	91-124
Number of pages	34
Journal	Mathematische Zeitschrift
Volume	262
Issue number	1
DOIs	https://doi.org/10.1007/s00209-008-0364-z
Publication status	Published - 2009

Keywords

Adjoint orbit
Classical Lie algebra
Compact Lie group
Conjugacy class
Orbital measure

ASJC Scopus subject areas

General Mathematics

Access to Document

10.1007/s00209-008-0364-z

Cite this

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abstract = "We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μk is singular to Lebesgue measure or μk ∈ L2. The minimum k for which μk ∈ L2 is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.",

keywords = "Adjoint orbit, Classical Lie algebra, Compact Lie group, Conjugacy class, Orbital measure",

author = "Gupta, {Sanjiv Kumar} and Hare, {Kathryn E.} and Sobhan Seyfaddini",

note = "Funding Information: S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. K. E. Hare was supported in part by NSERC.",

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AU - Gupta, Sanjiv Kumar

AU - Hare, Kathryn E.

AU - Seyfaddini, Sobhan

N1 - Funding Information: S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. K. E. Hare was supported in part by NSERC.

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N2 - We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μk is singular to Lebesgue measure or μk ∈ L2. The minimum k for which μk ∈ L2 is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.

AB - We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μk is singular to Lebesgue measure or μk ∈ L2. The minimum k for which μk ∈ L2 is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.

KW - Adjoint orbit

KW - Classical Lie algebra

KW - Compact Lie group

KW - Conjugacy class

KW - Orbital measure

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