
 Transformation Groups and Mathematical Physics
Fifth meeting, December 56, Hamburg University, see the permanent website for previous meetings. This meeting is supported by the Center for Mathematical Physics and the DFG Priority Programme 1388 Darstellungstheorie.
Local organisers:
Ingo Runkel,
Christoph Schweigert,
Christoph Wockel

Accommodation
Please contact Birgit Mehrabadi for accommodation and questions about reimbursement.
Some accomodation at reasonable prices is avaible at the DESY hostel which is at a 40 minutes ride from the mathematics department. Mention "Center for Mathematical Physics" when booking.
Directions
The talks will take place in the seminar room to the Klima Campus, Grindelberg 5, near to the math department. It can easily be reached by taking the subway U2 or U3 (both lines also stop at the central station) to the station Schlump and then follow the street "Beim Schlump" (which starts at the station, so the direction is unique) to its end. Then turn left, it is the second or third building on the left hand side (unfortunately, googlemaps does not know "Grindelberg 5", the buliding is roughly where googlemaps would find "Grindelberg 1").
Enlarge this view
Schedule
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 Kinvariant 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 HarishChandra 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 selfadjoint 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, L2norms, 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 Kfinite vectors.
This is joint work with T. Kobayashi, G. Mano, and J. Möllers.
 Alexey Petukhov:
Connection between kspherical 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 kvariety.
I proove that if Gr(k, V) is a spherical variety then the variety P(V) is kspherical too and find out all pairs (k,(g,V))
such that the variety Gr(k, V) is spherical.

Henrik Seppaenen:
BorelWeil 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 KarlHermann Neeb.

Rafal Suszek:
Gerbes, symmetries and generalised geometry
The presence of the abelian gerbe
G in a rigourous formulation of the twodimensional conformal
field theory of the string on a metric space (M, g), prerequisite to a nonanomalous 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 2category
naturally associated with the gerbe.
In the talk, this general framework will be discussed, alongside some of its fieldtheoretic and
geometric applications. Time permitting, we shall illustrate our considerations with physically
relevant examples, to wit, the maximally symmetric defect quiver in the WessZuminoWitten
model, the gauged sigma model on a Gspace, and the Tduality construction.
