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Given a Hermitian symmetric space M of noncompact type, we show, among other things, that the metric compactification of M with respect to its Carathéodory distance is homeomorphic to a closed ball in its tangent space. We first give a complete description of the horofunctions in the compactification of M via the realisation of M as the open unit ball D of a Banach space (V , ‖ · ‖) equipped with a particular Jordan structure, called a JB ∗ -triple. We identify the horofunctions in the metric compactification of (V , ‖ · ‖) and relate its geometry and global topology, via a homeomorphism, to the closed unit ball of the dual space V ∗ . Finally, we show that the exponential map exp 0 : V ⟶ D at 0 ∈ D extends to a homeomorphism between the metric compactifications of (V , ‖ · ‖) and (D , ρ) , preserving the geometric structure, where ρ is the Carathéodory distance on D. Consequently, the metric compactification of M admits a concrete realisation as the closed dual unit ball of (V , ‖ · ‖) . [ABSTRACT FROM AUTHOR]

There is a known hyperkähler structure on any complexified Hermitian symmetric space G/K, whose construction relies on identifying G/K with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space Gu/K0. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on G/K; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on T ∗ G u / K 0 . We highlight the intermediate Kähler structures, which share a holomorphic action of G related to the one on G/K, but moment geometry related to that of T ∗ G u / K 0 . As an application, for the real form G0 ⊂ G corresponding to G0/K0, the Hermitian symmetric space of noncompact type, we give a strategy for study of the action on G/K using the moment-critical subsets for the intermediate structures. We give explicit computations for SL(2). [ABSTRACT FROM AUTHOR]

This paper deals with a limiting case motivated by contact geometry. The limiting case of a tensorial characterization of contact hypersurfaces in Kähler manifolds leads to Hopf hypersurfaces whose maximal complex subbundle of the tangent bundle is integrable. It is known that in non-flat complex space forms and in complex quadrics such real hypersurfaces do not exist, but the existence problem in other irreducible Kähler manifolds is open. In this paper we construct explicitly a one-parameter family of homogeneous Hopf hypersurfaces, whose maximal complex subbundle of the tangent bundle is integrable, in a Hermitian symmetric space of non-compact type and rank two. These are the first known examples of such real hypersurfaces in irreducible Kähler manifolds. [ABSTRACT FROM AUTHOR]

We introduce the symplectic group Sp 2 (A , σ) over a noncommutative algebra A with an anti-involution σ . We realize several classical Lie groups as Sp 2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups Sp 2 (A , σ) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of Sp 2 (A , σ) on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action. When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space X Sp 2 (A , σ) , and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp 2 (A , σ) ) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A. [ABSTRACT FROM AUTHOR]

We use loop group factorizations to construct Darboux transformations, Permutability formulas, Scaling Transformations, and explicit soliton solutions of the 1 -d Schrödinger flows on compact Hermitian symmetric spaces. [ABSTRACT FROM AUTHOR]

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space. [ABSTRACT FROM AUTHOR]

It is known that the cotangent bundle ΩY of an irreducible Hermitian symmetric space Y of compact type is stable. We show that if X ⊂ Y is a subvariety whose structure sheaf has a short split resolution and such that the restriction map Pic(Y ) → Pic(X) is surjective, then, apart from a few exceptions, the restriction ΩY∣X is a stable bundle. We then address some cases where the Picard group increases by restriction. [ABSTRACT FROM AUTHOR]

Let Ξ be the crown domain associated with a non-compact irreducible Hermitian symmetric space G / K. We give an explicit description of the unique G-invariant adapted hyper-Kähler structure on Ξ, i.e., compatible with the adapted complex structure Jad and with the G-invariant Kähler structure of G / K. We also compute invariant potentials of the involved Kähler metrics and the associated moment maps. [ABSTRACT FROM AUTHOR]

We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of order α+∞+···+∞ ( m times) and its subgroup K ≅ U(∞), which consists of block diagonal unitary matrices with the identity block of order α and a matrix u ∈ U(∞) repeated m times. It turns out that there is a natural multiplication on the space G// K of conjugacy classes. We construct 'spectral data' of conjugacy classes, which visualize the multiplication and are sufficient for reconstructing a conjugacy class. [ABSTRACT FROM AUTHOR]