Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

Sanjiv Kumar Gupta; Kathryn E. Hare

doi:10.1007/s00605-008-0029-3

Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

Sanjiv Kumar Gupta, Kathryn E. Hare

Mathematics & Statistics

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

4 Citations (Scopus)

Abstract

We prove that if μ_a = m_k*δ_a*m_K is the K-bi-invariant measure supported on the double coset K a K {succeeds or equal to} SU (n), for K = SO(n), then μ_a^k is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ_a, supported on the minimal dimension double coset has the property that μ_a^n-1 is singular to the Haar measure.

Original language	English
Pages (from-to)	27-43
Number of pages	17
Journal	Monatshefte fur Mathematik
Volume	159
Issue number	1
DOIs	https://doi.org/10.1007/s00605-008-0029-3
Publication status	Published - 2009

Keywords

Absolutely continuous measure
Bi-invariant measure
SU(n)
Symmetric space

ASJC Scopus subject areas

Mathematics(all)

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10.1007/s00605-008-0029-3

Cite this

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abstract = "We prove that if μa = mk*δa*mK is the K-bi-invariant measure supported on the double coset K a K {succeeds or equal to} SU (n), for K = SO(n), then μak is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μa, supported on the minimal dimension double coset has the property that μan-1 is singular to the Haar measure.",

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note = "Funding Information: This research was supported in part by grants from NSERC and Sultan Qaboos University. S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. The authors would also like to thank the referee for carefully reading the paper.",

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AU - Gupta, Sanjiv Kumar

AU - Hare, Kathryn E.

N1 - Funding Information: This research was supported in part by grants from NSERC and Sultan Qaboos University. S. K. Gupta appreciates the hospitality of the Department of Pure Mathematics at the University of Waterloo where some of this research was done. The authors would also like to thank the referee for carefully reading the paper.

PY - 2009

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N2 - We prove that if μa = mk*δa*mK is the K-bi-invariant measure supported on the double coset K a K {succeeds or equal to} SU (n), for K = SO(n), then μak is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μa, supported on the minimal dimension double coset has the property that μan-1 is singular to the Haar measure.

AB - We prove that if μa = mk*δa*mK is the K-bi-invariant measure supported on the double coset K a K {succeeds or equal to} SU (n), for K = SO(n), then μak is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μa, supported on the minimal dimension double coset has the property that μan-1 is singular to the Haar measure.

KW - Absolutely continuous measure

KW - Bi-invariant measure

KW - SU(n)

KW - Symmetric space

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Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

Abstract

Keywords

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