On the product formula on non-compact Grassmannians

Piotr Graczyk; Patrice Sawyer

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arXiv · 1212.0002

On the product formula on non-compact Grassmannians

Piotr Graczyk·Patrice Sawyer

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Abstract

We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star \delta_{e^Y}^\natural$ of two orbital measures on the symmetric space $SO_0(p,q)/SO(p)\timesSO(q)$, $q>p$. We prove sharp conditions on $X$, $Y\in\a$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for $\SO_0(p,q)/\SO(p)\times\SO(q)$ will also serve for the spaces $SU(p,q)/S(U(p)\timesU(q))$ and $Sp(p,q)/Sp(p)\timesSp(q)$, $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $\delta_{e^X}^\natural$.

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Piotr Graczyk, Patrice Sawyer. 2012-11-30. On the product formula on non-compact Grassmannians. https://arxiv.org/abs/1212.0002

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