Research Seminar "Lie Theory"
The Research Seminar usually meets on Thursday 4-6 pm in E2.304
Next
talk: Thursday, February 4,
2010, 4:15-5:15 pm, in E2.304 Multiplier Theorems for the Jacobi Transform |
| Abstract: The
Jacobi transform is a generalization of the spherical transform on a
rank
one Riemannian symmetric space, where the root multiplicities are
replaced by
two complex numbers (whose real parts are usually taken to be greater
than
-1/2). By generalizing the Stanton-Tomas expansion for spherical
functions to
the setting of Jacobi functions, we have established multiplier theorems
of
Hörmander-type for the Jacobi transform that give sufficient conditions
for a
function to be a multiplier. These generalize all known results for the
spherical transform on a rank one space. The outstanding, and still
open,
problem is to characterize such multipliers.
|
Program for the winter term 2009/2010 (October through March)
Thursday, October 15, 2009; 4:15-5:45 pm in E2.304
J. Hilgert (Paderborn): Lie Supergroups
Abstract: In this talk I will give the definition of a Lie Supergroup and describe how Lie supergroups are related to Lie groups via the functor of points approach.
Thursday, October 22, 2009; 4:15-5:45 pm in E2.304
A. Alldridge (Paderborn): The supergroup pairs approach to Lie supergroups IAbstract: We introduce supergroup pairs. Such a pair is a linear action of a Lie group on a Lie superalgebra which extends the adjoint action. We show that any supergroup pair defines a Lie supergroup, and that this gives rise to an equivalence of categories.
Thursday, October 29, 2009; 4:15-5:45 pm in E2.304
A. Alldridge (Paderborn): The supergroup pairs approach to Lie supergroups IIAbstract: We introduce supergroup pairs. Such a pair is a linear action of a Lie group on a Lie superalgebra which extends the adjoint action. We show that any supergroup pair defines a Lie supergroup, and that this gives rise to an equivalence of categories.
Thursday, November 5, 2009; 4:15-5:45 pm in E2.304
H. Biebinger (Paderborn): Lie supergroup actions and homogeneous superspaces 1Abstract: We show that for a super Lie group G and a closed subsupergroup there exists a "quotient" super manifold G/H which is a homogeneous space. We show that actions of a Lie supergroup on an arbitrary supermanifold are in natural bijection with actions of the supergroup pair of the Lie supergroup. As an example, we consider the super projective space as a homogeneous space.
Thursday, November 12, 2009; 4:15-5:45 pm in E2.304
H. Biebinger (Paderborn): Lie supergroup actions and homogeneous superspaces 2Abstract: We show that for a super Lie group G and a closed subsupergroup there exists a "quotient" super manifold G/H which is a homogeneous space. We show that actions of a Lie supergroup on an arbitrary supermanifold are in natural bijection with actions of the supergroup pair of the Lie supergroup. As an example, we consider the super projective space as a homogeneous space.
Thursday, November 19, 2009; 4:15-5:45 pm in E2.304
W. Palzer (Paderborn): Integration on Supermanifolds
Abstract: In the first part of this talk the Berezin bundle Ber will be introduced. This enables us to give a suitable definition of integration on compactly supported sections of Ber. In the second part we take a look on the approach made by Rothstein to get over the limitation on compactness.
Thursday, November 26, 2009; 4:15-5:45 pm in E2.304
M. Laubinger (Paderborn): Typical Representations of Classical Lie Superalgebras
Abstract: In the 1970ies, Kac defined and classified classical Lie superalgebras. He then defined typical highest weight modules and gave twelve equivalent characterizations of typicality. In this talk, we will motivate the notion of typicality by presenting some of the equivalent characterizations.
Transformation Groups and Mathematical Physics
Fifth meeting, December 5-6, Hamburg University,Thursday, December 10, 2009; 4:15-5:45 pm in E2.304
M. Laubinger (Paderborn): Representations of infinite-dimensional Orthogonal and Symplectic Groups.
Abstract: We discuss the bosonic and fermionic Fock space associated with a separable Hilbert space H, and then introduce the CAR and CCR algebras, which play a role in quantum mechanics. Lastly, we show how to obtain representations of the restricted orthoghonal and symplectic group of H.Thursday, December 17, 2009; 4:15-5:45 pm in E2.304
S. Hansen (Paderborn): Ruelle Resonances for Anosov Diffeomorphisms
Abstract: This is a report about a paper by Faure, Roy and Sjöstrand (2008) who used microlocal analysis to study the spectral properties of the transfer operator of an Anosov diffeomorphism. Resonances occur as the eigenvalues of associated non-selfadjoint spectral problems. A calculus of pseudo-differential operators of variable order is (explained and) used.
Thursday, January 7, 2010; 4:15-5:15 pm in E2.304
J. Emonds (Aachen): Strictly Positive Definite Functions on the Torus and Other Compact GroupsAbstract:
A rich literature and rather complete information is available on positive definite functions on (locally) compact groups. By contrast, little is known about strictly positive definite functions. Already the question whether such functions exist is non-trivial and closely related toe the groups topology. Starting with positive definite functions we describe how to approach strictly positive definite functions for compact resp. compact abelian groups. For tori we provide a coomplete characterization via the Fourier transform.
Thursday, January 28, 2010; 4:15-5:45 pm in E2.304
W. Palzer (Paderborn): Integration on Supermanifolds II - Transformation behaviour
Abstract: In the classical case the integral of a volume form is independent from the choice of a coordinate system but on superdomains the value of the integral changes for some coordinate changes. The aim of this talk is to figure out a general transformation formula for coordinate changes and to classify the coordinates which lead to the same integral value. This classification will enable us to define an integral on supermanifolds for Berezin forms which do not vanish on the boundary.
Thursday, Ferbuary 4, 2010; 4:15-5:15 pm in E2.304
T. Johansen (Kiel): Multiplier Theorems for the Jacobi TransformAbstract: The Jacobi transform is a
generalization of the spherical transform on a rank
one Riemannian symmetric space, where the root multiplicities are
replaced by
two complex numbers (whose real parts are usually taken to be greater
than
-1/2). By generalizing the Stanton-Tomas expansion for spherical
functions to
the setting of Jacobi functions, we have established multiplier theorems
of
Hörmander-type for the Jacobi transform that give sufficient conditions
for a
function to be a multiplier. These generalize all known results for the
spherical transform on a rank one space. The outstanding, and still
open,
problem is to characterize such multipliers.
By a suitable choice of
complex parameters, we also obtain results for Damek-Ricci spaces (where
some of
the results are known in the literature but not always accompanied with
proofs),
for the Heckman-Opdam transform of a rank one root system with a
complex-valued
multiplicity function (where the results were previously unknown), as
well as
for the spherical Laplace transform on the noncompactly causal symmetric
spaces
SO(1,n)/SO(1,n-1) (improving previous results).
Program of previous semesters
summer 2009, winter 2008/2009 , summer 2008 , winter 2007/2008, summer 2007 , winter 2006/2007, summer 2006, winter 2005/2006, summer 2005, winter 2004/2005, summer 2004
Please send questions, remarks or suggestions to Joachim Hilgert