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THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

Published online by Cambridge University Press:  26 April 2021

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

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).

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc

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Footnotes

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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