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CONVOLUTION OF ORBITAL MEASURES IN SYMMETRIC SPACES

Part of: Abstract harmonic analysis Global differential geometry

Published online by Cambridge University Press:  09 February 2011

BOUDJEMÂA ANCHOUCHE  and
SANJIV KUMAR GUPTA
BOUDJEMÂA ANCHOUCHE
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, PO Box 36, Al-Khoud 123, Muscat, Sultanate of Oman (email: anchouch@squ.edu.om)
SANJIV KUMAR GUPTA*
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, PO Box 36, Al-Khoud 123, Muscat, Sultanate of Oman (email: gupta@squ.edu.om)
*
✉For correspondence; e-mail: gupta@squ.edu.om
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Abstract

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Let G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

Keywords

convolutiondouble cosetorbital measuressymmetric spaces

MSC classification

Secondary: 43A80: Analysis on other specific Lie groups 53C35: Symmetric spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 470 - 485
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The authors are grateful to Sultan Qaboos University for its support.

References

[1]Dunkl, C., ‘Operators and harmonic analysis on the sphere’, Trans. Amer. Math. Soc. 125 (1966), 250–263.Google Scholar
[2]Gupta, S. and Hare, K., ‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419.Google Scholar
[3]Gupta, S. and Hare, K., ‘Singularity of orbits in classical Lie algebras’, Geom. Funct. Anal. 13 (2003), 815–844.CrossRefGoogle Scholar
[4]Gupta, S. and Hare, K., ‘Convolutions of generic orbital measures in compact symmetric spaces’, Bull. Aust. Math. Soc. 79(3) (2009), 513–522.CrossRefGoogle Scholar
[5]Gupta, S. and Hare, K., ‘L 2-singular dichotomy for orbital measures of classical compact Lie groups’, Adv. Math. 222 (2009), 1521–1573.Google Scholar
[6]Gupta, S. and Hare, K., ‘Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)’, Monatsh. Math. 159 (2010), 27–43.Google Scholar
[7]Gupta, S., Hare, K. and Seyfaddini, S., ‘L 2-singular dichotomy for orbital measures of classical simple Lie algebras’, Math. Z. 262 (2009), 91–124.Google Scholar
[8]Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, New York, 1978).Google Scholar
[9]Knapp, A. W., Lie Groups: Beyond an Introduction (Birkhäuser, Boston, 2002).Google Scholar
[10]Mimura, M. and Toda, H., Topology of Lie Groups, Translations of Mathematical Monographs, 91 (American Mathematical Society, Providence, RI, 1991).Google Scholar
[11]Ragozin, D., ‘Central measures on compact simple Lie groups’, J. Funct. Anal. 10 (1972), 212–229.Google Scholar
[12]Ragozin, D., ‘Zonal measure algebras on isotropy irreducible homogeneous spaces’, J. Funct. Anal. 17 (1974), 355–376.Google Scholar
[13]Varadarajan, V. S., Lie Groups and Lie Algebras and their Representations (Springer, New York, 1984).Google Scholar