Cycle Connectivity and Automorphism Groups of Flag Domains

A flag domain $D$ is an open orbit of a real form $G_0$ in a flag manifold $Z=G/P$ of its complexification. If $D$ is holomorphically convex, then, since it is a product of a Hermitian symmetric space of bounded type and a compact flag manifold, ${Aut}(D)$ is easily described. If $D$ is not holomorphically convex, then in our previous work (American J. Math, 136, Nr.2 (2013) 291-310 (arXiv: 1003.5974)) it was shown that ${Aut}(D)$ is a Lie group whose connected component at the identity agrees with $G_0$ except possibly in situations which arise in Onishchik's list of flag manifolds where ${Aut}(Z)^0$ is larger than $G$. These exceptions are handled in detail here. In addition substantially simpler proofs of some of our previous work are given.Comment: To appear in Birkh\"auser Progress Reports "Current Developments and Retrospectives in Lie Theor

On hyperbolicity of SU(2)-equivariant, punctured disc bundles over the complex affine quadric

Given a holomorphic line bundle over the complex affine quadric $Q^2$, we investigate its Stein, SU(2)-equivariant disc bundles. Up to equivariant biholomorphism, these are all contained in a maximal one, say $\Omega_{max}$. By removing the zero section to $\Omega_{max}$ one obtains the unique Stein, SU(2)-equivariant, punctured disc bundle over $Q^2$ which contains entire curves. All other such punctured disc bundles are shown to be Kobayashi hyperbolic.Comment: 15 pages, v2: minor changes, to appear in Transformation Group

Symmetry classes of disordered fermions

Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(V+V*), a bilinear form on V+V* which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.Comment: 52 pages, dedicated to Freeman J. Dyson on the occasion of his 80th birthda