Minisymposium Mathematics Cluster GQT

Geometry & Quantum Theory

Organizer & Chair: Klaas Landsman (Radboud Universiteit Nijmegen)
Waaier 4, 14:15 - 16:15

Extensions of tempered representations

Eric Opdam (Universiteit van Amsterdam)

The spaces of higher extensions in the abelian category of smooth representations of a reductive p-adic group between two irreducible tempered representations can be computed explicitly in terms of so-called analytic R-groups. We will explain some of the techniques on which this computation is based. The result has surprising applications in harmonic analysis. For example it implies Kazhdan’s orthogonality conjecture for elliptic inner products of admissible characters of reductive p-adic groups.

(joint work with Maarten Solleveld)

A support theorem for the horospherical transform on a semisimple symmetric space

Job Kuit (Universiteit Utrecht)

A symmetric space is a pseudo-Riemannian manifold with the property that in each point x the geodesic reflection in x extends to a globally defined isometry. Let X be a Riemannian symmetric space of the non-compact type, i.e. a symmetric space with a Riemannian structure and negative sectional curvature. The connected component of the isometry group G acts transitively on X and the stabilizer of a given point is a maximal compact subgroup K. Therefore X is isometric to the homogeneous space G∕K. An example of a Riemannian symmetric space of non-compact type is the Poincardisk D SL(2,R)SO(2).

There exists a distinguished family of submanifolds of X called horospheres. (A horosphere in a symmetric space G∕K is an orbit in X of the unipotent radical of a minimal parabolic subgroup of G.) The horospheres in the Poincardisk D are the circles tangential to the boundary of the disk. (If N denotes the subgroup of SL(2,R) of upper triangular matrices with 1’s on the diagonal, then the horospheres are the orbits of subgroups conjugate to N.)

The horospherical transform Rφ of a suitable function φ on X is the function on the set of horospheres Ξ given by

R φ(ξ) =   φ(x)d x    (ξ ∈ Ξ).
         ξ     ξ
Here dξx denotes the measure on a horosphere ξ induced by the Riemannian structure on X. The horospherical transform is an example of a Radon transform.

In 1973 Helgason proved the following support theorem: Let V be a closed ball in X. If Rφ(ξ) = 0 for every horosphere ξ with ξ V = , then φ(x) = 0 for x∕∈V .

We present a support theorem for the horospherical transform on a general pseudo-Riemannian semisimple symmetric space. Our theorem generalizes the theorem by Helgason.

An example of the geometric Langlands correspondence

Jochen Heinloth (Universiteit van Amsterdam)

The relation between analytic properties of modular forms and arithmetic results has led to many famous results and conjectures. A geometric analogue of this conjectural relation is called geometric Langlands correspondence. We will try to give an idea of what modular forms are in this context.

Unlike in the classical theory of modular forms, in this geometric version very few explicit examples of modular forms are known. In joint work with B.C. Ngô and Z. Yun – which was motivated by work of Gross and Frenkel - we found an explicit series of such forms which on the one hand give an example of the (wild) geometric Langlands correspondence and on the other hand turn out to be closely related to classical Kloosterman sums.

On a new Witt group

Michael Mueger (Radboud Universiteit Nijmegen)

Let k be a field. Two non-degenerate quadratic forms on finite dimensional k-vector spaces are called Witt equivalent if they become isomorphic upon adding suitable hyperbolic quadratic forms. The set of equivalence classes is the Witt group W(k) of k. Similarly, one can consider quadratic forms over free abelian groups or over finite abelian groups. The Witt group Wf over finite abelian groups was computed by C. T. C. Wall and others. We give an interpretation, due to V. Drinfeld, S. Gelaki, D. Nikshych and V. Ostrik, of Wf in terms of certain braided tensor categories. Enlarging the class of categories under consideration to all modular categories, where the role of hyperbolic quadratic forms is taken over by the center construction of Majid, Drinfeld, Joyal and Street, leads to a new Witt group WM that contains Wf as a proper subgroup. WM is of considerable interest in mathematical physics. We state what is presently known about WM as well as a conjecture. This is joint work with A.Davydov, D.Nikshych and V.Ostrik, cf. arXiv:1009.2117.