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
gupta@squ.edu.om
Kathryn E. Hare
Dept. of Pure Mathematics, University of Waterloo, Canada
kehare@uwaterloo.ca
[Abstract-pdf]
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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