On the product formula on non-compact Grassmannians

TL;DR

This paper establishes sharp conditions for the absolute continuity of convolutions of orbital measures on certain non-compact Grassmannian symmetric spaces, with implications for product formulas of spherical functions.

Contribution

It provides the first precise criteria for the absolute continuity of orbital measure convolutions on these spaces, extending to multiple types of symmetric spaces.

Findings

Sharp conditions for absolute continuity of convolutions on SO(p,q)/SO(p)×SO(q)

Criteria applicable to SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q)

Results on absolute continuity of convolution powers of orbital measures

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 $\delta_{e^X}^\natural$.