Soji Kaneyuki

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.

On the automorphism group of the generalized conformal structure of a symmetric R-space

Abstract In this paper we define a canonical locally flat generalized conformal structure on a symmetric R -space of the rank greater than 1. We prove that the group of automorphisms of this structure coincides with the noncompact group of automorphisms of the symmetric space.

Signatures of roots and a new characterization of causal symmetric spaces

Let g be a real semisimple Lie algebra, let τ be a Cartan involution of g. Choose a maximal abelian subspace a in the (−l)-eigenspace of τ, and let △ be the root system for (g, a). Oshima-Sekiguchi [11] introduced the notion of a signature function ∈ on △, which is a map of △∪(0) to {±1} satisfying a multiplicative property. Using ∈ and τ, they define a new involution △∈ of g by putting △∈(X) = ∈(a)τ(X), where X is a root vector for a root a ∈ △ ∪ (0). The resulting symmetric pair (a, τ∈) is said to be a symmetric pair of type K ∈. A (ℤ)-graded Lie algebra (or shortly GLA) \(g = \sum _{k = - v}^v{g_k}\) is said to be of the v-th kind, if g −1 ≠ (0) and g ±v ≠ (0). In [7, 6], we have worked out the classification and construction of real semisimple GLA’s.

The Laplace-Beltrami Operator in Various Coordinates

When the hyperball R ℝ(m, n) is a bounded domain in ℝ m x n , the L-B operator associated to the metric (1.58) in rectangular coordinates X=(x jα ) is known to be of the form (1.63). Now we try to express the L-B operator in matrix polar coordinates.

Semisimple Graded Lie Algebras

Let \(\mathfrak{g}\) be a real semisimple Lie algebra and τ be a Cartan involution of \(\mathfrak{g}\) and let \(\mathfrak{g} = \mathfrak{k} + \mathfrak{p}\) be the Cartan decomposition by τ, where \(\tau {{|}_{\mathfrak{k}}} = 1\) and \(\tau {{|}_{\mathfrak{p}}} = - 1\). Let a be \(\mathfrak{a}\) maximal abelian subspace of \(\mathfrak{p}\) and \(\mathfrak{h}\) be a Cartan subalgebra of \(\mathfrak{g}\) containing \(\mathfrak{a}\). Then we have \(\mathfrak{h} = {{\mathfrak{h}}^{ + }} + \mathfrak{a}\), where \({{\mathfrak{h}}^{ + }} = \mathfrak{h} \cap \mathfrak{k}\) and \(\mathfrak{a} = \mathfrak{h} \cap \mathfrak{p}\). Let \({{\mathfrak{g}}^{\mathbb{C}}}\) and \({{\mathfrak{h}}^{\mathbb{C}}}\) be the complexifications of \(\mathfrak{g}\) and \(\mathfrak{h}\). Then \({{\mathfrak{h}}^{\mathbb{C}}}\) is a Cartan subalgebra of \({{\mathfrak{g}}^{\mathbb{C}}}\). Let \(\tilde{\Delta } = \Delta ({{\mathfrak{g}}^{\mathbb{C}}},{{\mathfrak{h}}^{\mathbb{C}}})\) be the root system for the pair left \(({{\mathfrak{g}}^{\mathbb{C}}},{{\mathfrak{h}}^{\mathbb{C}}})\). If we put \({{\mathfrak{h}}_{\mathbb{R}}} = i{{\mathfrak{h}}^{ + }} + \mathfrak{a}\) then any root is real-valued on the real subspace \({{\mathfrak{h}}_{\mathbb{R}}}\) of \({{\mathfrak{h}}^{\mathbb{C}}}\). Since the Killing form B of \(\mathfrak{g}\) is positive-definite on \({{\mathfrak{h}}_{\mathbb{R}}}\), a root \(\alpha \in \tilde{\Delta }\) can be viewed as an element of \({{\mathfrak{h}}_{\mathbb{R}}}\). We have thus \(\tilde{\Delta } \subset {{\mathfrak{h}}_{\mathbb{R}}}\). Let σ be the conjugation of \({{\mathfrak{g}}^{\mathbb{C}}}\) with respect to \(\mathfrak{g}\) . Then \(\sigma {{|}_{\mathfrak{a}}} = 1\) and \(\sigma {{|}_{{i{{\mathfrak{h}}^{ + }}}}} = - 1\), and hence σ leaves \({{\mathfrak{h}}_{\mathbb{R}}}\) stable. Therefore σ permutes roots in \(\tilde{\Delta }\). Let us put \({{\tilde{\Delta }}_{ \bullet }} = \tilde{\Delta } \cap i{{\mathfrak{h}}^{ + }}\), the set of imaginary roots with respect to \(\mathfrak{h}\). We then have \({{\tilde{\Delta }}_{ \bullet }} = \{ \alpha \in \tilde{\Delta }:\sigma (\alpha ) = - \alpha \}\). A lexicographic order > on \(\tilde{\Delta }\) is called a σ-order, if σis order-preserving on \(\tilde{\Delta } - {{\tilde{\Delta }}_{ \bullet }}\), or σ (α)> 0, as long as α > 0, \(\alpha \in \tilde{\Delta } - {{\tilde{\Delta }}_{ \bullet }}\). Such an order is given by choosing a basis \(\{ {{H}_{1}}, \ldots {{H}_{r}},{{H}_{{r + 1}}}, \ldots ,{{H}_{l}}\}\) of \({{\mathfrak{h}}_{\mathbb{R}}}\) such that \(\{ {{H}_{1}}, \ldots ,{{H}_{r}}\}\) is a basis of \(\mathfrak{a}\). Now let us fix a σ-order in \(\tilde{\Delta }\). The simple root system \(\tilde{\prod } = \{ {{\alpha }_{1}}, \ldots ,{{\alpha }_{l}}\}\) of \(\tilde{\Delta }\) with respect to this σ-order is called aσ -fundamental system of \(\tilde{\Delta }\). The subset \({{\tilde{\prod }}_{ \bullet }} = \tilde{\prod } \cap {{\tilde{\Delta }}_{ \bullet }}\) of \(\tilde{\prod }\) is a basis for \({{\tilde{\Delta }}_{ \bullet }}\).

Symmetric R-Spaces

Let \( g = \sum\nolimits_{k = - v}^v {gk} \)be a real simple GLA of the v-th kind, and (Z, T) be the associated pair. Let G 0 be the group of grade-preserving automorphisms of g. Note that G 0 coincides with the centralizer C(Z)of Z in Aut g, and that Lie G 0 =g0.Let G be the open subgroup of Aut g generated by G 0 and the adjoint group of g: G= G 0Adg,Let U = G 0 exp(g1 + ··· + gv), which is a parabolic subgroup of G.

Pseudo-Hermitian Symmetric Spaces

Let G/H be an almost effective symmetric coset space of a connected Lie group G. We do not assume H to be compact. If G is simple, G/H is called a simple symmetric space. If the linear isotropy representation of H is irreducible (resp. reducible), then G/H is called simple irreducible (resp. reducible). If G/H admits a G-invariant complex structure J and a G-invariant pseudo-Hermitian metric (with respect to J), then a G/H is called pseudo-Hermitian. Simple symmetric spaces were classified infinitesimally by Berger [1].