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}
}
• Published 30 November 2012
• Mathematics
• arXiv: Representation Theory
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 $SU(p,q)/S(U(p… 7 Citations Tables from this paper The smoothness of convolutions of orbital measures on complex Grassmannian symmetric spaces • Mathematics • 2019 It is well known that if$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},$is CONVOLUTION OF ORBITAL MEASURES ON SYMMETRIC SPACES OF TYPE$C_{p}$AND$D_{p}$• Mathematics Journal of the Australian Mathematical Society • 2014 Abstract We study the absolute continuity of the convolution${\it\delta}_{e^{X}}^{\natural }\star {\it\delta}_{e^{Y}}^{\natural }$of two orbital measures on the symmetric spaces The absolute continuity of convolution products of orbital measures in exceptional symmetric spaces • Mathematics • 2015 Let G be a non-compact group, K the compact subgroup fixed by a Cartan involution and assume G / K is an exceptional, symmetric space, one of Cartan type E, F or G. We find the minimal integer, L(G), The Absolute continuity of convolutions of orbital measures in SO(2n+1) Let G be a compact Lie group of Lie type Bn, such as SO(2n+1). We characterize the tuples (x1, ..., xL) of the elements xj ∈ G which have the property that the product of their conjugacy classes has A Laplace-Type Representation of the Generalized Spherical Functions Associated with the Root Systems of Type A In this paper, we extend the iterative expression for the generalized spherical functions associated with the root systems of type A previously obtained (Sawyer in Trans Am Math Soc 349(9):3569–3584, Convolution of orbital measures on symmetric spaces: a survey • Mathematics • 2014 This survey summarizes a long and fruitful collaboration of the authors on the properties of the convolutions of orbital measures on symmetric spaces or, equivalently, on the product formula for References SHOWING 1-10 OF 20 REFERENCES Some Convexity Results for the Cartan Decomposition • Mathematics Canadian Journal of Mathematics • 2003 Abstract In this paper, we consider the set$\text{S}=a\left( {{e}^{X}}K{{e}^{Y}} \right)$where$a\left( g \right)$is the abelian part in the Cartan decomposition of$g$. This is exactly the A sharp criterion for the existence of the density in the product formula on symmetric spaces of type A n • Mathematics • 2010 In this paper, we find sharp conditions on X, Y ∈ a for the existence of the density of the measure δ eX ? δ eY intervening in the product formula for the spherical functions on the symmetric spaces Zonal measure algebras on isotropy irreducible homogeneous spaces This paper analyzes the convolution algebra M(K\GK) of zonal measures on a Lie group G, with compact subgroup K, primarily for the case when M(K\GK) is commutative and GK is isotropy irreducible. A Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces • Mathematics • 2010 We study the absolute continuity of the measures δeX1♮⋆⋯⋆δeXm♮ and of (δeX♮)⋆l on the Riemannian symmetric spaces X of noncompact type for nonzero elements Xj, X∈a. For m,l⩾r+1, where r is the rank The Central Limit Theorem on SO0(p, q)/SO(p)×SO(q) We obtain a central limit theorem for the space SO0(p, q)/SO(p)×SO(q). To achieve this, we derive a Taylor expansion of the spherical function on the group SO0(p, q). Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman–Opdam hypergeometric functions of type BC. For specific discrete series Positive convolution structure for a class of Heckman–Opdam hypergeometric functions of type BC Abstract In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman–Opdam hypergeometric functions of type BC. For specific Spherical Functions on SO0(p, q)/ SO(p) × SO(q) • P. Sawyer • Mathematics Canadian Mathematical Bulletin • 1999 Abstract An integral formula is derived for the spherical functions on the symmetric space${G}/{K\,=\,{\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times
The product formula for the spherical functions on symmetric spaces of noncompact type.
• Mathematics
• 2003
In this paper, we prove the existence of the product formula for the spherical functions on symmetric spaces of noncompact type. To this end, we study the analyticity properties of the Cartan
Geometric Analysis on Symmetric Spaces
A duality in integral geometry A duality for symmetric spaces The fourier transform on a symmetric space The Radon transform on $X$ and on $X_o$. Range questions Differential equations on symmetric