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Algèbres de Volterra et transformation de Laplace sphérique sur certains espaces symétriques ordonnés. (Volterra algebras and spherical Laplace transform on certain ordered symmetric spaces). (French) Zbl 0656.43003
Harmonic analysis, symmetric spaces and probability theory, Cortona/Italy 1984, Symp. Math. 29, 183-196 (1987).
[For the entire collection see Zbl 0633.00012.]
In this paper the author is interested in the symmetric spaces $$X=G\ell (m,{\mathbb{F}})/U(p,q:{\mathbb{F}})$$ where $${\mathbb{F}}={\mathbb{R}}$$ or $${\mathbb{C}}$$. On these spaces there is an invariant order which allows to define an algebra of Volterra kernels and a spherical Laplace transform that generalizes the usual Laplace transform.
The invariant order on X is inherited from $${\mathcal H}_ m({\mathbb{F}})$$ the space of symmetric, resp. hermitian matrices with real, resp. complex coefficients. The order is defined by the cone of positive definite matrices $$\Omega ={\mathcal P}_ m({\mathbb{F}})$$ and X is “identified” as an orbit of $$G=G\ell (m,{\mathbb{F}}):$$ $$X=\rho (G)J$$, $$J=\left( \begin{matrix} I_ p\\ 0\end{matrix} \begin{matrix} 0\\ -I_ q\end{matrix} \right)$$. A Volterra kernel is a function k(x,g) on $$X\times X$$ that is continuous on $$\Gamma =\{(x,g)|$$ $$x\geq y\}$$ and zero outside $$\Gamma$$. For such kernels multiplication and invariance are defined in the paper. A theorem says that the algebra of invariant Volterra kernels is commutative. In the second and main part of the paper a family of spherical functions is defined and computed in terms of an Iwasawa decomposition. This family makes possible to define Laplace transform for invariant Volterra kernels. For this Laplace transform a multiplication theorem is proved.
Reviewer: W.Kugler

##### MSC:
 43A32 Other transforms and operators of Fourier type 53C35 Differential geometry of symmetric spaces 22E30 Analysis on real and complex Lie groups