Laplace transform on the cone of positive-definite matrices The title says most. Let $P_p$ be the cone of positive-definite $p \times p$ matrices. 
One can define the Laplace transform of (the distribution of) a random matrix with values in $P_p$ by (for instance, Muirhead: "Aspects of Multivariate Analysis") the integral
\begin{equation}
     \phi(\Theta) = \int_{P_p} \exp(\sum_{j\le k}^p \theta_{jk} a_{jk}) f(A)\; dA
\end{equation}
where $f(A)$ is the density function of $A$. (And $\Theta$ is a symmetric $p\times p$-matrix)
So the question is: I am searching for references for this Laplace transform, inversion theorems, numerical methods, known transform formulas, .... etc ???
Thanks for answers!  Now, I am searching for those references, but one is really difficult to find, that is, volume 2 of Audrey Terras' book: "Harmonic analysis on symmetric spaces II"
I have found the first volume, but volume II cannot even be found on Springers own website!  Any ideas about how to find it?
 A: Second editions of both volumes are about to be published.
It will probably take a year to update them though.
              audrey
A: This is more like a comment...
I am also interested in this question.
My interest comes from the paper:
Kostant, B.; Sahi, S. (1991), "The Capelli Identity, tube domains, and the generalized Laplace transform", Advances in Math. 87: 71–92, doi:10.1016/0001-8708(91)90062-C
As far as I understand cones of positive matrices are very related to Jordan algebras
and there is a comprehensive book (I have file of it, can send you), which in particular contains many things on Laplace transform:
Analysis on symmetric cones
Jacques Faraut, Adam Korányi - 1994 - 382 pages
Oxford University Press, Oxford, UK
A: Laplace and Fourier transforms on the cone of positive definite matrices are somewhat well-studied. The framework is, as already mentioned by Alexander, that of "analysis on symmetric spaces." Here are some references where this subject is developed in greater detail:


*

*Harmonic analysis on symmetric spaces and applications, vols. I and II (particularly chapter 4 for vol. II covers positive matrices), by A. Terras, Springer Verlag. I think Terras also lists a short table of Fourier transforms and their inverses; those should help with Laplace transforms too.

*Analysis on symmetric cones, by J. Faraut, A. Koranyi, Clarendon Press, 1994. In particular, have a look at Chapter VII

*Geometric analysis on symmetric spaces, S. Helgason, AMS, 1991.(Chapter III discusses the Fourier transform)


For computation, it seems you have to do eigenvector decomposition, and numerical integration thereafter. On a related note, I mention the following toolbox by P. Koev:
Hypergeometric function of a matrix argument
I mention this because these functions are closely related to the Laplace transform that you have mentioned (as detailed out in Muirhead's book).
A: I know this isn't an answer but I don't have enough rep to comment. I too am interested in the question. @Alexander Chervov: That book sounds interesting, I'd appreciate it if you could email me a copy (if you don't mind of course). My email address is tarmetjoe@yahoo.com. Thanks.
