Siegel Domain Research Articles

Special algebraic properties of Käehler algebras

Homogeneous Kahler manifolds M are frequently investigated via Kahler algebras (g, k, j, p), where g denotes a Lie algebra of infinitesimal automorphisms of M and k the isotropy subalgebra of some point in M. Moreover, / corresponds to the complex structure tensor and p to the Kahler form. In particular, Kahler algebras have been used intensively in the proof of the geometric Fundamental Conjecture for homogeneous Kahler manifolds: Every homogeneous Kahler manifold is a holomorphic fiberbundle over a homogeneous bounded domain in which the fiberis (with the induced Kahler metric) the product of a flat homogeneous Kahler manifold and a compact simply connected homogeneous Kahler manifold. Two additional properties of Kahler algebras have proven to be particularly useful. One is that g or ad g is an algebraic Lie algebra. The second one is the assumption that p is the differentialof a leftinvariant 1-form, p=d<o. This is the case of /-algebras. It has been investigated intensively by Gindikin, Piatetskii-Shapiro,Vinberg and others. The proof of the Fundamental Conjecture for homogenous Kahler manifolds is much shorter for /-algebras than for general Kahler algebras. This is due to some extent to the fact that one can embed a /-algebra into an algebraic /-algebra. The purpose of this note is threefold. First we want to prove that for the Lie algebra gM of all infinitesimal automorphisms of an arbitrary homogeneous Kahler manifold M, the Lie algebra z.dgM is algebraic. Secondly, we decompose gM into the orthogonal sum of /-invariant subalgebras. This decomposition will be of importance for a forthcoming publicationin which we give a detailed description of kM and the Kahler form p. The orthogonal decomposition in question has a simple geometric interpretation. It is essentially induced by a representation of the base domain (occuring in the Fundamental Conjecture) as a Siegel domain of type three.

Discrete subgroups of algebraic groups over local fields of positive characteristics

It is shown in this paper that ifG is the group ofk-points of a semisimple algebraic groupG over a local fieldk of positive characteristic such that all itsk-simple factors are ofk-rank 1 and Γ ⊂G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that ifG has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.

FINITE-DIMENSIONALITY OF THE GROUP OF CR AUTOMORPHISMS OF A STANDARD CR MANIFOLD, AND PROPER HOLOMORPHIC MAPPINGS OF SIEGEL DOMAINS

It is shown that local CR automorphisms of nondegenerate standard CR manifolds are rational and form a finite-dimensional Lie group. It is established that proper holomorphic mappings of nondegenerate Siegel domains are biholomorphic and rational. Bibliography: 14 titles.

Homogeneous K�hler manifolds andT-algebras inN=2 supergravity and superstrings

Motivated by the problem of the moduli space of superconformal theories, we classify all the (normal) homogeneous Kahler spaces which are allowed in the coupling of vector multiplets toN=2 SUGRA. Such homogeneous spaces are in one-to-one correspondence with the homogeneous quaternionic spaces (≠ℍH n ) found by Alekseevskii. There are two infinite families of homogeneous non-symmetric spaces, each labelled by two integers. We construct explicitly the corresponding supergravity models. They are described by acubic functionF, as in flat-potential models. They are Kahler-Einstein if and only if they are symmetric. We describe in detail the geometry of the relevant manifolds. They are Siegel (bounded) domains of the first type. We discuss the physical relevance of this class of bounded domains for string theory and the moduli geometry. Finally, we introduce theT-algebraic formalism of Vinberg to describe in an efficient way the geometry of these manifolds. The homogeneous spaces allowed inN=2 SUGRA are associated to rank 3T-algebras in exactly the same way as the symmetric spaces are related to Jordan algebras. We characterize theT-algebras allowed inN=2 supergravity. They are those for which theungraded determinant is a polynomial in the matrix entries. The Kahler potential is simply minus the logarithm of this “naive” determinant.

A description of a space of holomorphic discrete series by means of the Fourier transform on the Šilov boundary

Realizing holomorphic discrete series representations on Hilbert spaces H (Λ) of vector valued holomorphic functions on a Siegel domain of type II, we establish a Paley-Wiener type theorem for H (Λ) analogous to the classical one for Hardy space on the upper half plane.

Lie group structures and reproducing kernels on homogeneous siegel domains

We introduce a continuous family of Lie group structures on a homogeneous Siegel domain. We associate to each one a Hilbert space of holomorphic functions getting an interpolation between the Bergman and the Szego kernels.

Flat totally geodesic submanifolds of quasisymmetric Siegel domains

Flat totally geodesic submanifolds of quasisymmetric Siegel domains

Analytic Extension of the Holomorphic Discrete Series

Introduction. The embedding R ' naturally divides into two half-planes, each with R as boundary. These domains together with their function theory play an important role in the harmonic analysis of R. As but one example there is the theorem of Paley and Wiener that describes any square integrable function on R as a sum of boundary values of holomorphic functions (in Hardy spaces) on these domains. Much of the theory has natural extensions to R, to tube domains in C based on cones, and even to more general Siegel domains. However, the dominant influence is abelian harmonic analysis. On the other hand, if G is a semisimple Lie group over R it may often be embedded in its natural complexification Gc. Does any of the abelian

The Dual of the Bergman Space A 1 in Symmetric Siegel Domains of Type II

The Dual of the Bergman Space A 1 in Symmetric Siegel Domains of Type II

The dual of the Bergman space 𝐴¹ in symmetric Siegel domains of type 𝐼𝐼

An affirmative answer is given to the following conjecture of R. Coifman and R. Rochberg: in any symmetric Siegel domain of type II, the dual of the Bergman space A 1 {A^1} coincides with the Bloch space of holomorphic functions and can be realized as the Bergman projection of L ∞ {L^\infty } .

Simply transitive groups and Kähler structures on homogeneous Siegel domains

We determine the Lie algebras of all simply transitive groups of automorphisms of a homogeneous Siegel domain D D as modifications of standard normal j j -algebras. We show that the Lie algebra of all automorphisms of D D is a "complete isometry algebra in standard position". This implies that D D carries a riemannian metric g ~ \tilde g with nonpositive sectional curvature satisfying Lie Iso ⁡ ( D , g ~ ) = Lie Aut D \operatorname {Iso}(D,\tilde g) = \operatorname {Lie}\; \operatorname {Aut}\, \text {D} . We determine all Kähler metrics f f on D D for which the group Aut ⁡ ( D , f ) \operatorname {Aut}(D,f) of holomorphic isometries acts transitively. We prove that in this case Aut ⁡ ( D , f ) \operatorname {Aut}(D,f) contains a simply transitive split solvable subgroup. The results of this paper are used to prove the fundamental conjecture for homogeneous Kähler manifolds admitting a solvable transitive group of automorphisms.

The transition to chaos for a special solution of the area-preserving quadratic map

Following previous work on chaotic boundaries of half-plane Hamiltonian maps a special solution of the area-preserving quadratic map is introduced and investigated. The breakdown of regular bounded motion on invariant curves is found from the radius of convergence of a power series whose successive terms oscillate wildly due to the presence of small divisors. Previous techniques for taming such series are found to be insufficient and new ones are introduced. It is found that half-plane Hamiltonian maps appear to have certain universal features and that the chaotic boundary has similarities to the boundaries of Siegel domains of complex conformal maps. The chaotic boundary function α c(ν) has some interesting new features which are not fully understood.

On half-homogeneous hyperbolic manifolds and Siegel domains

On half-homogeneous hyperbolic manifolds and Siegel domains

Local characterization of holomorphic automorphisms of Siegel domains

Local characterization of holomorphic automorphisms of Siegel domains

Renormalization group analysis of quasi-periodicity in analytic maps

Analytic maps of the form\(f(z) = e^{2\pi i\Omega } z + \mathcal{O}(z^2 )\) display quasiperiodicity when Ω satisfies a diophantine condition. Quasiperiodic motion is confined to a neighborhood of the origin known as a Siegel domain. The boundary of this domain obeys universal scaling relations. In this paper we investigate these scaling relations through a renormalization group analysis, and we discuss singularities and asymptotic form of the scaling function.

Homogeneous Siegel domains

In 1935 E. Cartan classified all symmetric bounded domains [6]. At that time he proved that a bounded symmetric domain is homogeneous with respect to its group of holomorphic automorphisms. Thus the more general problem of investigating homogeneous bounded domains arose. It was known to E. Cartan that all homogeneous bounded domains of dimension ≤3 are symmetric [6]. For domains of higher dimension little was known. The first example of a 4-dimensional, homogeneous, non-symmetric bounded domain was provided by I. Piatetsky-Shapiro [41]. In several papers he investigated homogeneous bounded domains [20], [21], [41], [42], [43]. One of the main results is that all such domains have an unbounded realization of a certain type, as a so-called Siegel domain. But many questions still remained open. Amongst them the question for the structure and explicit form of the infinitesimal automorphisms of a homogeneous Siegel domain.

Siegel Domains and Representations of Jordan Algebras

Siegel Domains and Representations of Jordan Algebras

Siegel domains and representations of Jordan algebras

In the analysis of infinitesimal automorphisms of arbitrary Siegel domains, a certain class of nonsemisimple Jordan algebras occurs. The description of all the infinitesimal automorphisms of the domain may be based on a study of representations of the associated Jordan algebra satisfying a certain "strange identity". In this paper, all the possibilities for the Jordan algebra and representations satisfying the identity are given.

On the equivalence problem for a certain class of generalized Siegel domains. II

On the equivalence problem for a certain class of generalized Siegel domains. II

The solution of the $$\overline \partial $$ -Neumann problem in a strictly pseudoconvex Siegel domain

In this paper, we solve the O-Neumann problem on (0,q) forms, q > 0, in the strictly pseudoconvex Siegel domain. The metric we use is invariant under the action of the Heisenberg group on the domain. We give a formula for a kernel of the Neumann operator, the fundamental solution of the problem, in (1.6) and Theorem 4.2, and we prove uniqueness of the Neumann operator. As far as we know, this is the first example of an explicit solution of the ~-Neumann problem. Greiner and Stein [GS] constructed a parametrix for the problem on (0, 1) forms in bounded strictly pseudoconvex domains. Phong [P] has announced an explicit construction of a parametric on (03) forms in the Siegel domain in C "+ 1, n > 1. Harvey and Polking [HP] have since constructed a Neumann kernel for the ~-Neumann problem on (p, q) forms on the ball. We obtain the kernel of the Neumann operator by integrating the fundamental solution of the corresponding heat equation ([S1], [$2]) with respect to time. To carry out the integration, we require estimates on the fundamental solution for large time. We did not need such estimates in [$1] and [$2]. Also, to prove that the resulting kernel is locally integrable, we require more precise estimates for small time. In Sect. 1, after some preliminaries, we state our main theorem. We then summarize the results of [S1 ] and [$2]. We solve the corresponding elliptic problem on C" + 1 in Sect. 2. In Sect. 3 we study the operator which corrects the boundary values of the solution we obtained in Sect. 2. We solve the O-Neumann problem in Sect. 4. Most of the work for this paper was done while I was a member of the Institute for Advanced Study. I wish to thank the Institute for its hospitality.