Jacques Faraut

Analysis and geometry on complex homogeneous domains

Part 1 Function spaces on complex semi-groups, Jacques Faraut: Hilbert spaces of holomorphic functions invariant cones and complex semi-groups positive unitary representations Hilbert function spaces on complex semi-groups Hilbert function spaces on SL(2,C) Hilbert function spaces on a complex semi-simple Lie group. Part 2 Graded Lie algebras and pseudo-hermitian symmetric spaces, Soji Kaneyuki: semi-simple graded Lie algebras symmetric R-spaces pseudo-hermitian symmetric spaces. Part 3 Function spaces on bounded symmetric domains, Adam Koranyi: Bergman kernel and Bergman metric symmetric domains and symmetric spaces construction of the hermitian symmetric spaces structure of symmetric domains the weighted Bergman spaces differential operators function spaces. Part 4 The heat kernels of non-compact symmetric spaces, Qi-keng Lu: introduction the Laplace-Beltrami operator in various co-ordinates the integral transformations the heat kernel of the hyperball Rr(m,n) the harmonic forms on the complex Grassmann manifold the horo-hypercircle coordinate of a complex hyperball the heat kernel of R11(m) the matrix representation of NIRGSS. Part 5 Jordan triple systems, Guy Ross: polynomial identities Jordan algebras the quasi-inverse the generic minimal polynomial tripotents and Pierce decomposition hermitian positive JTS further results and open problems. References.

Pseudo-Hermitian symmetric spaces of tube type

Let Ω be an open connected cone in a real vector space V ≃ ℝn. One defines G(Ω) = {g ∈ GL(n,∝) ∣ gΩ = Ω}. The cone Ω is said to be homogeneous if the group G(Ω) acts transitively on Ω. For the beginning let us assume that Ω is convex and that \(\bar \Omega \) is pointed (this means that \(\bar \Omega \) ∩ (}\(\bar \Omega \)) = {0}). The convex cone Ω is said to be selfdual if there exists a positive inner product on V such that Ω✶ = Ω, where the open dual cone Ω✶ is defined by

Projections of orbital measures for the action of a pseudo-unitary group

G(\Omega ) = \{ g \in GL(n,\mathbb{R}|g\Omega = \Omega \} .

The Laplace-Beltrami Operator in Various Coordinates

The open convex cone Ω is said to be symmetric if it is homogenous and selfdaul. Let us recall the connection between symmetric convex cones and Jordan algebras. A Jordan algebra V is a vector space equipped with a product, i.e., a bilinear map V × V → V such that (J1) xy = xy, (J2) x(x 2 ) = x 2 (xy).

Invariant Cones in a Lie Algebra and Complex Semi-groups

Let G be a linear Lie group, and \( \Gamma \left( C \right) \subset {G^\mathbb{C}} \) be a complex semi-group. We will study Hilbert spaces \( \mathcal{H} \subset \mathcal{O}\left( {\Gamma \left( {{C^0}} \right)} \right) \) which are G × G- invariant, for the action defined by

Hilbert Function Spaces on a Complex Olshanski Semi-group in a Complex Semi-simple Lie Group

f\left( \gamma \right) \mapsto f\left( {g_2^{ - 1}\gamma {g_1}} \right)