# On the product formula on non-compact Grassmannians

@article{Graczyk2012OnTP, title={On the product formula on non-compact Grassmannians}, author={Piotr Graczyk and Patrice Sawyer}, journal={arXiv: Representation Theory}, year={2012} }

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…

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## 7 Citations

The smoothness of convolutions of orbital measures on complex Grassmannian symmetric spaces

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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…

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Abstract We study the absolute continuity of the convolution ${\it\delta}_{e^{X}}^{\natural }\star {\it\delta}_{e^{Y}}^{\natural }$ of two orbital measures on the symmetric spaces…

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In this paper, we extend the iterative expression for the generalized spherical functions associated with the root systems of type A previously obtained (Sawyer in Trans Am Math Soc 349(9):3569–3584,…

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This survey summarizes a long and fruitful collaboration of the authors on the properties of the convolutions of orbital measures on symmetric spaces or, equivalently, on the product formula for…

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