We study the absolute continuity of the convolution of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ 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 SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to the study of absolute continuity of convolution powers of an orbital measure .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-1,
author = {Piotr Graczyk and Patrice Sawyer},
title = {On the product formula on noncompact Grassmannians},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {145-167},
zbl = {1286.43012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-1}
}
Piotr Graczyk; Patrice Sawyer. On the product formula on noncompact Grassmannians. Colloquium Mathematicae, Tome 131 (2013) pp. 145-167. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-2-1/