Automorphism groups of causal symmetric spaces of Cayley type and bounded symmetric domains

Symmetric spaces of Cayley type are a higher dimensional analogue of a one-sheeted hyperboloid in R3. They form an important class of causal symmetric spaces. To a symmetric space of Cayley type M, one can associate a bounded symmetric domain of tube type D. We determine the full causal automorphism group of M. This clarifies the relation between the causal automorphism group and the holomorphic automorphism group of D.     

Weakly symmetric spaces and bounded symmetric domains

LetG be a connected, simply-connected, real semisimple Lie group andK a maximal compactly embedded subgroup ofG such thatD=G/K is a hermitian symmetric space. Consider the principal fiber bundleM=G/K _s G/K, whereK _s is the semisimple part ofK=K _s ·Z _K ⁰ andZ _K ⁰ is the connected center ofK. The natural action ofG onM extends to an action ofG ¹=G×Z _K ⁰ . We prove as the main result thatM is weakly symmetric with respect toG ¹ and complex conjugation. In the case whereD is an irreducible classical bounded symmetric domain andG is a classical matrix Lie group under a suitable quotient, we provide an explicit construction ofM=D×S ¹ and determine a one-parameter family of Riemannian metrics onM invariant underG ¹. Furthermore,M is irreducible with respect to . As a result, this provides new examples of weakly symmetric spaces that are nonsymmetric, including those already discovered by Selberg (cf. [M]) for the symplectic case and Berndt and Vanhecke [BV1] for the rank-one case.Research partially supported by an NSF grant. The author wishes to thank the International Erwin Schroedinger Institute for its hospitality during the preparation of this paper.     

Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

Bloch spaces on bounded symmetric domains in complex Banach spaces

We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

D p spaces on bounded symmetric domains ofC n

In this paper, the spaceD ^p(Ω) of functions holomorphic on bounded symmetric domain ofC ⁿ is defined. We prove thatH ^p(Ω)⊂D ^p(Ω) if 0<p≤2, andD ^p(Ω)⊂H ^p(Ω) ifp≥2, and both the inclusions are proper. Further, we find that some theorems onH ^p(Ω) can be extended to a wider classD ^p(Ω) for 0<p≤2.     

Automorphism groups of nilpotent groups and spaces

A pair of finitely generated, torsion-free nilpotent groups G₁,G₂ is constructed with the properties that G₁ and G₂ are p-isomorphic for all primes p, yet Aut(G₁) and Aut(G₂) are not isomorphic. The example constructed is compared to an analogous example in the homotopy category of simply connected, finite CW-complexes.     

A characterization of bounded symmetric domains of type IV

Let ?? be a bounded symmetric domain of type IV and dimension bigger than four. We show that a Stein manifold of the same dimension as ?? and with the same automorphism group is biholomorphic to ??.     

Weighted composition operators on weighted Bergman spaces of bounded symmetric domains

In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator W _φψ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.     

Weighted composition operators between different Bergman spaces of bounded symmetric domains

In this paper, we consider the boundedness and compactness of the weighted composition operators between different Bergman spaces of bounded symmetric domains in terms of the Carleson measure. As an application, we study the multipliers between different Bergman spaces.     

Automorphism groups for measurable spaces

In [1], the notion of a rigid separable measurable space was defined, and such spaces were shown to exist. We expand upon this idea and ask what are the possible automorphism groups for such spaces. We show that there is only one such non-trivial group which is Abelian (a countable product of two element groups), and that this group is realised as the automorphism group of some separable space. A particular class of such spaces is characterised in terms of rigid components.Finally, an example of a measurable space which is rigid in the strict sense is constructed. This answers a question of K.P.S. Bhaskara Rao and B.V. Rao.     

Automorphism group of exceptional symmetric domains RVI

Here we give the definition of the exceptional symmetric Siegel domain R_VI(27) in C²⁷, and compute the exceptional symmetric domain ?_VI(27) = τ(R_VI(27)), where t is the Bergman mapping of the Siegel domainR _VI (27). Moreover, we present the holomorphical automorphism group Aut (?_VI(27)) of the exceptional symmetric domain (?_VI(27)).     

Automorphism groups of Cayley graphs generated by block transpositions and regular Cayley maps

This paper deals with the Cayley graph $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$, where the generating set consists of all block transpositions. A motivation for the study of these particular Cayley graphs comes from current research in Bioinformatics. As the main result, we prove that $\text{Aut}\left(\mathrm{Cay}({\mathrm{Sym}}_n,T_n)\right)$ is the product of the left translation group and a dihedral group ${\mathsf{D}}_{n+1}$ of order $2(n+1)$. The proof uses several properties of the subgraph $\Gamma$ of $\mathrm{Cay}({\mathrm{Sym}}_n,T_n)$ induced by the set $T_n$. In particular, $\Gamma$ is a $2(n?2)$-regular graph whose automorphism group is ${\mathsf{D}}_{n+1},$ $\Gamma$ has as many as $n+1$ maximal cliques of size $2$, and its subgraph $\Gamma \left(V\right)$ whose vertices are those in these cliques is a 3-regular, Hamiltonian, and vertex-transitive graph. A relation of the unique cyclic subgroup of ${\mathsf{D}}_{n+1}$ of order $n+1$ with regular Cayley maps on ${\text{Sym}}_n$ is also discussed. It is shown that the product of the left translation group and the latter group can be obtained as the automorphism group of a non-$t$-balanced regular Cayley map on ${\text{Sym}}_n$.     

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns of non-compact Riemannian symmetric spaces . The basic result is that affine symmetric spaces of can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

     

Schatten class composition operators on weighted Bergman spaces of bounded symmetric domains

Summary We obtain trace ideal criteria for 0A ² () of a Bounded symmetric diomain in _n.     

Automorphism group of exceptional symmetric domains RV

The exceptional symmetric Siegel domain RV(16) in C16 is defined. The exceptional classical domain (R)v(16)=τ(RV(16)) is computed, where τ is the Bergman mapping of the Siegel domain RV(16). And holomorphical automorphism group Aut (RV(16)) of the exceptional symmetric Siegel domain RV(16) is presented.     

Bloch constants of bounded symmetric domains

Let and be two irreducible bounded symmetric domains in the complex spaces and respectively. Let be the Euclidean metric on and the Bergman metric on . The Bloch constant is defined to be the supremum of , taken over all the holomorphic functions and , and nonzero vectors . We find the constants for all the irreducible bounded symmetric domains and . As a special case we answer an open question of Cohen and Colonna.

     

Compressions of infinite-dimensional bounded symmetric domains

\noindent If D is a bounded symmetric domain in a complex Banach space Z, then the identity component G of its group of biholomorphic automorphisms permits a natural embedding into a complex Banach—Lie group H acting partially on Z. A typical model is the action of the group PSL(2,C) by Moebius transformations. In this paper we show that the interior S ⁰ of the compression semigroup S := { h ∈ H: h.D \subeq D } has a polar decomposition in the sense that S ⁰ = G \exp(W_G^0), where W_G \subeq ig is a closed convex invariant cone and the polar map G \times W_G^0 → S^0 is a diffeomorphism. March 31, 2000