Some Convexity Results for the Cartan Decomposition

2003 ◽  
Vol 55 (5) ◽  
pp. 1000-1018 ◽  
Author(s):  
P. Graczyk ◽  
P. Sawyer
Keyword(s):  
Symmetric Spaces ◽  
Singular Values ◽  
Product Formula ◽  
Spherical Functions ◽  
Cartan Decomposition ◽  
Simple Description ◽  
Noncompact Type

AbstractIn this paper, we consider the set = a(eXKeY) where a(g) is the abelian part in the Cartan decomposition of g. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of SL(3; F) where F= R, Cor H. In particular, we show that is convex.We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.

scholarly journals CONVOLUTION OF ORBITAL MEASURES ON SYMMETRIC SPACES OF TYPE AND

2014 ◽  
Vol 98 (2) ◽  
pp. 232-256 ◽  
Author(s):  
P. GRACZYK ◽  
P. SAWYER
Keyword(s):  
Absolute Continuity ◽  
Symmetric Spaces ◽  
Product Formula ◽  
Spherical Functions ◽  
The Absolute

AbstractWe 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 $\mathbf{SO}_{0}(p,p)/\mathbf{SO}(p)\times \mathbf{SO}(p)$, $\mathbf{SU}(p,p)/\mathbf{S}(\mathbf{U}(p)\times \mathbf{U}(p))$ and $\mathbf{Sp}(p,p)/\mathbf{Sp}(p)\times \mathbf{Sp}(p)$. We prove sharp conditions on $X$, $Y\in \mathfrak{a}$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.

Scattering theory for symmetric spaces of noncompact type

1993 ◽  
Vol 72 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Ralph S. Phillips ◽  
Mehrdad M. Shahshahani
Keyword(s):  
Scattering Theory ◽  
Symmetric Spaces ◽  
Noncompact Type
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