### Jordan algebras and generalized principle series representations.

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The group SU(1,d) acts naturally on the Hilbert space $L\xb2\left(Bd{\mu}_{\alpha}\right)(\alpha >-1)$, where B is the unit ball of ${\u2102}^{d}$ and $d{\mu}_{\alpha}$ the weighted measure ${(1-|z\left|\xb2\right)}^{\alpha}dm\left(z\right)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...

We prove a central limit theorem for certain invariant random variables on the symmetric cone in a formally real Jordan algebra. This extends form the previous results of Richards and Terras on the cone of real positive definite $n\times n$ matrices.

We give a complete description of the boundary behaviour of the generalized hypergeometric functions, introduced by Faraut and Koranyi, on Cartan domains of rank 2. The main tool is a new integral representation for some spherical polynomials, which may be of independent interest.

We study the relative discrete series of the ${L}^{2}$-space of the sections of a line bundle over a bounded symmetric domain. We prove that all the discrete series appear as irreducible submodules of the tensor product of a holomorphic discrete series with a finite dimensional representation.

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