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Schedule for Saturday 5 December

Schedule for Sunday 6 December

Abstracts

• Alexander Alldridge Harmonic analysis on symmetric superspaces
  
  In the study of large N statistics of random matrix ensembles, embeddings of all of Cartan's ten infinite series of Riemannian symmetric spaces G/K into complex symmetric superspaces occur in a natural fashion. This plays a role in applications to mesoscopic physics (e.g. in the work of Wegner, Efetov, and Zirnbauer).
  In this context, questions of harmonic analysis (such as to the validity of a spherical Fourier inversion theorem for K-invariant functions) arise. This suggests a systematic study of harmonic analysis on symmetric superspaces.
  In an ongoing programme with J. Hilgert (Paderborn) and M.R. Zirnbauer (Köln) we are developing the harmonic analysis of invariant differential operators and spherical functions in this context.
  In the talk, I shall explain the framework of our investigation and present our results on the generalisation Chevalley's restriction theorem, invariant Berezin integration, and their application to a generalisation of the Harish-Chandra isomorphism. Many new features occur: For instance, the variety defined by the algebra of symmetric invariants (which is affine the classical case) may be singular.
  
  
• Joachim Hilgert: Special functions associated with minimal representations of O(p,q)
  
  We develop a theory of "special functions" associated to a certain fourth order differential operator Dμ,ν on R depending on two parameters μ,ν. For integers μ,ν≥-1 with μ+ν∈2N0 this operator extends to a self-adjoint operator on L2(R+,xμ+ν+1 dx) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, L2-norms, integral representations and various recurrence relations.
  
  The fourth order differential operator Dμ,ν arises as the radial part of the Casimir action in the Schrödinger model of the minimal representation of the group O(p,q), and our "special functions" give K-finite vectors.
  
  This is joint work with T. Kobayashi, G. Mano, and J. Möllers.
  
  
• Alexey Petukhov: Connection between k-spherical Grassmanians and spherical (g,k)-modules
  
  Famous article of A.Beilinson and J.Bernstein explains that there is a close connection between modules of a Lie algebra g and a geometry of a maximal Grassmanian of g - the variety of all Borel subalgebras. Using this connection I'm able to solve a problem introduced by Ivan Penkov and Vera Serganova - classify all pairs (sln, k) such that there exists a spherical (sln, k)-module.
  Theorem. There exists a spherical (sl(V), k)-module if and only if there exists k such that Gr(k, V) is a spherical k-variety. I proove that if Gr(k, V) is a spherical variety then the variety P(V) is k-spherical too and find out all pairs (k,(g,V)) such that the variety Gr(k, V) is spherical.
  
  
• Henrik Seppaenen: Borel-Weil theory for root graded Lie groups over commutative Banach algebras
  
  Root graded Lie groups are generalizations of semisimple complex Lie groups which are allowed to be infinite dimensional. For a root graded Lie group G one can define parabolic subgroups. Any holomorphic Banach representation of a parabolic subgroup, P, defines a holomorphic (Banach) vector bundle over the complex manifold G/P. In particular, a holomorphic character of P defines a line bundle. We give a characterization of those line bundles which admit nonzero global holomorphic sections in the case when G/P is a scalar extension of a compact flag variety by a commutative Banach algebra.
  
  This is joint work with Karl-Hermann Neeb.
  
  
• Rafal Suszek: Gerbes, symmetries and generalised geometry
  
  The presence of the abelian gerbe G in a rigourous formulation of the two-dimensional conformal field theory of the string on a metric space (M, g), prerequisite to a non-anomalous realisation of the conformal symmetry in the quantised theory, significantly modifies the structure of internal symmetries of the theory in that it puts them in correspondence with Killing sections of the generalised tangent bundle, rather than Killing vector fields, over the target space M of the string. The study of a symplectic presentation of these symmetries, formulated in terms of target- space structures (M, g, G ) transgressed to the field space of the theory, leads to the emergence of a general framework of description of stringy symmetries and dualities that uses a 2-category naturally associated with the gerbe. In the talk, this general framework will be discussed, alongside some of its field-theoretic and geometric applications. Time permitting, we shall illustrate our considerations with physically relevant examples, to wit, the maximally symmetric defect quiver in the Wess-Zumino-Witten model, the gauged sigma model on a G-space, and the T-duality construction.