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THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

Part of: Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  26 April 2021

SANJIV KUMAR GUPTA  and
KATHRYN E. HARE
SANJIV KUMAR GUPTA
Affiliation:
Dept. of Mathematics, Sultan Qaboos University, P.O. Box 36, Muscat, Al Khodh 123, Sultanate of Oman e-mail: gupta@squ.edu.om
KATHRYN E. HARE*
Affiliation:
Dept. of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
*
e-mail: kehare@uwaterloo.ca
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Abstract

Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$. For the special case of the orbital measures, $\nu _{a_{i}}$, supported on the double cosets $Ka_{i}K$, where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

Keywords

orbital measurespherical functionsymmetric space

MSC classification

Primary: 43A90: Spherical functions
Secondary: 43A85: Analysis on homogeneous spaces 22E30: Analysis on real and complex Lie groups

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 2 , October 2022 , pp. 188 - 207
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Robert Yuncken

This research is supported in part by NSERC 2016-03719 and by Sultan Qaboos University. The authors thank Acadia University for their hospitality when this research was done.

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