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Journal of Lie Theory 31 (2021), No. 2, 335--349
Copyright Heldermann Verlag 2021

The Smoothness of Convolutions of Singular Orbital Measures on Complex Grassmannians

Sanjiv Kumar Gupta
Dept. of Mathematics, Sultan Qaboos University, Sultanate of Oman

Kathryn E. Hare
Dept. of Pure Mathematics, University of Waterloo, Canada


It is well known that if $G/K$ is any irreducible symmetric space and $\mu_{a}$ is a continuous orbital measure supported on the double coset $KaK$, then the convolution product, $\mu _{a}^{k},$ is absolutely continuous for some suitably large number $k\leq \dim G/K$. The minimal value of $k$ is known in some symmetric spaces and in the special case of compact groups or rank one compact symmetric spaces it has even been shown that $\mu _{a}^{k}$ belongs to the smaller space $L^{2}$ for some $k$. Here we prove that this $L^{2}$ property holds for all the compact, complex Grassmannian symmetric spaces, $SU(p+q)/S(U(p)\times U(q))$. Moreover, for the orbital measures at a dense set of points $a$, we prove that $\mu _{a}^{2}\in L^{2}$ (or $\mu_{a}^{3}\in L^{2}$ if $p=q$).

Keywords: Orbital measure, spherical functions, complex Grassmannian symmetric space, absolute continuity.

MSC: 43A90, 43A85; 58C35, 33C50.

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