Journal of Lie Theory

no. 2
The Smoothness of Convolutions of Singular Orbital Measures on Complex Grassmannians
Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 335-349
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 $p=q$).
DOI: 10.5802/jolt.1174
Classification: 43A90, 43A85, 58C35, 33C50
Keywords: Orbital measure, spherical functions, complex Grassmannian symmetric space, absolute continuity 
@article{JOLT_2021_31_2_a2,
     author = {S. K. Gupta and K. E. Hare},
     title = {The {Smoothness} of {Convolutions} of {Singular} {Orbital} {Measures} on {Complex} {Grassmannians}},
     journal = {Journal of Lie Theory},
     pages = {335--349},
     year = {2021},
     volume = {31},
     number = {2},
     doi = {10.5802/jolt.1174},
     zbl = {1479.43014},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1174/}
}
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S. K. Gupta; K. E. Hare. The Smoothness of Convolutions of Singular Orbital Measures on Complex Grassmannians. Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 335-349. doi: 10.5802/jolt.1174

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