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CONVOLUTION OF ORBITAL MEASURES IN SYMMETRIC SPACES

Published online by Cambridge University Press:  09 February 2011

BOUDJEMÂA ANCHOUCHE
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, PO Box 36, Al-Khoud 123, Muscat, Sultanate of Oman (email: [email protected])
SANJIV KUMAR GUPTA*
Affiliation:
Department of Mathematics and Statistics, Sultan Qaboos University, PO Box 36, Al-Khoud 123, Muscat, Sultanate of Oman (email: [email protected])
*
For correspondence; e-mail: [email protected]
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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

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

References

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