Lie Group and Representation Theory Seminar, Kyoto 2005
Lie Group and Representation Theory Seminar, Kyoto 2005 | Date: | December 20 (Tue), 2005, 16:30-17:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Joachim Hilgert (Paderborn) |
| Title: | Hecke Operators on Spaces of Period Functions |
| Abstract: [pdf] |
Period functions are the analog of period polynomials, wellknown in the theory of modular forms, for Maass cusp forms. Before they were considered in the context of Maass forms they came up in physicists work on the transfer operator of geodesic flows on hyperbolic surfaces. The structure of these operators suggested the existence of Hecke type operators on spaces of period functions. It turns out that the constructions given in the context of transfer operators can be related to the standard Hecke operators on Maass forms in a precise way. |
| Date: | November 29 (Tue), 2005, 16:30-17:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Chifune Kai (Kyoto University) |
| Title: | A Characterization of symmetric Siegel domains by convexity of Cayley transform images |
| Abstract: [pdf] |
A homogeneous Siegel domain is a higher dimensional analogue of the
right (or upper) half plane, and is mapped to a bounded domain by the
Cayley transform. Among homogeneous Siegel domains, we have an important
subclass consisting of symmetric ones, which we characterize in this
talk using the parametrized family of Cayley transforms defined by
Nomura. This family includes the Cayley transforms associated with the
Bergman kernel and Szegö kernel, and if the domain is symmetric,
(the inverse of) the Cayley transform introduced by Korányi and Wolf.
In this talk, we show that the Cayley transform image is convex if and only if the domain is symmetric and the parameter is a specific one so that the Cayley transform coincides with the Kor\'anyi-Wolf Cayley transform. |
| Date: | November 22 (Tue), 2005, 16:30-17:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Simon Gindikin (Rutgers) |
| Title: | Holomorphic language for Cauchy-Riemann cohomology and representations |
| Abstract: [pdf] |
It turns out that in many explicit computations with $\bar \partial$- cohomlogy it is unconvenient to use by the traditional languages of Cech or Dolbeault. It was developped a conception of smoothly parameterized Cech cohomolgy, different versions of which were developpped in the collaboration with Eastwood and Wang. One of possibilities is to build a purely holomorphic language. I will give a review of constructions and several illustrating examples basically from theory of reprentations. |
| Date: | November 11 (Fri), 2005, 10:30-11:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Simon Gindikin (Rutgers) |
| Title: | Integral geometry and representations |
| Abstract: [pdf] |
The old project of Gelfand was to obtain the harmonic analysis on
symmetric and other homogeneous spaces using the horospherical
transform. The realization of
this program had an enormous obstruction since the initial version
of the horospherical transform annulates discrete series of representations.
Several years ago I suggested a modified version - horospherical
Cauchy transform - which is nontrivial at least for some discrete series.
I want to consider several examples , starting of SL(2,R), and to discuss hoe realistic looks today this Gelfand's project. |
| Date: | November 8 (Tue), 2005, 18:00-19:30 NOTE: The time schedule has been changed. |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Ivan Cherednik (University of North Carolina) |
| Title: | On unitary representations of rational dahas |
| Abstract: [pdf] |
The talk will be an attempt to show the importance of the analytic aspects of the rational dahas via the the simplest case. I will begin with discussing the eigenvalue problem for the Dunkl operator (one x), and then consider in full detail a simple example, that may be viewed as a starting point of the generaly theory of unitary representations of the rational dahas. This example is directly connected with the harmonic analysis on SL(2,R) and OSP(2|1); Dunkl operator is a square root of the radial part of the Laplace operator in this case, a kind of the Dirac operator. |
| Date: | November 1 (Tue), 2005, 16:30-17:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Jun O'Hara (Tokyo Metropolitan University) |
| Title: | Conformal geometry of curves |
| Abstract: [pdf] |
We study the space S(q,n) of (q+2)-dimensional vector subspaces of the
(n+2)-dimensional Minkowski space which intersect the light cone transversely.
It is a subset of an indefinite Grassmann manifold.
This space can be identified with the space of q-spheres in S^n.
I will explain the notion of a pencil, which is one-parameter family of
codimension 1 spheres in S^k.
Using pencils, I will give a pseudoorthonormal basis S(q,n).
The pseudo-Riemannian structure of S(q,n) allows us to give an interpretation of the ``infinitesimal cross ratio", which is a complex valued 2-form on the two point configuraion space of a knot K, KxK-(diagonal): The real part of it can be interpreted as an area element of a surface in S(q,n). |
| Date: | October 25 (Tue), 2005, 13:10-13:55 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Misha Pevzner (Reims) |
| Title: | Rankin-Cohen brackets and quantaziation of causal symmetric spaces |
| Abstract: [pdf] |
We consider semi-simple real Lie groups G such that the associated
Riemannian symmetric space G/K is a Hermitian symmetric space of
tube type and the non-Riemannian one G/H is a para-hermitian
symmetric space. Such symmetric spaces are usually called
causal symmetric spaces of Cayley type.
The first requirement implies that the Lie group G has
holomorphic discrete series representations acting on the space of
square integrable holomorphic functions on G/K. The fact that
G/H is para-hermitian, i.e. has a G-invariant splitting of the
tangent bundle into two isomorphic sub-bundles, allows us to build up
on G/H a symbolic calculus. It turns out that one can define a
G-covariant symbolic calculus on G/H generalizing the so-called
convolution first symbolic calculus on {\mathbb R}^2.
In the present talk we discuss two different ring structures on the set of holomorphic discrete series. First one comes from the convolution of functions on the symmetric cone underlying the Hermitian symmetric space of tube type G/K. The second one is non-commutative and is induced by the composition of operators whose symbols belong to the discrete series representations of the causal symmetric space of Cayley type G/H. We also discuss the relationship that exists between these ring structures, intertwining operators for tensor products of holomorphic discrete series representations and the so-called Rankin-Cohen brackets. |
| Date: | October 21 (Fri), 2005, 10:30-11:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Misha Pevzner (Reims) |
| Title: | Kontsevich Quantization and Duflo Isomorphism |
| Abstract: [pdf] |
Since the fundamental results by Chevalley, Harish-Chandra and
Dixmier one knows that
the set of ad-invariant polynomials on the dual of a Lie algebra of a
particular type (solvable, simple or nilpotent)
is isomorphic, as an algebra, to the center of the enveloping algebra. This
fact was generalized to an arbitrary finite-dimensional
real Lie algebra by M. Duflo in late 70's. His proof is based on the
Kirillov's orbits method that parametrizes infinitesimal characters
of unitary irreducible representations of the corresponding Lie group in
terms of co-adjoint orbits.
The Kontsevich' Formality theorem implies not only the existence of the Duflo map but shows that it is canonical. We shall describe this construction and indicate how does this construction extend to the whole Poisson cohomology of an arbitrary finite-dimensional real Lie algebra. |
| Date: | October 11 (Tue), 2005, 16:30-17:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Andreas Nilsson (RIMS) |
| Title: | Characterization of discrete Riesz transforms |
| Abstract: [pdf] |
The Riesz transforms in R^n have been characterized to be the only translation invariant linear operators satisfying some condtion of relative invariance under the conformal group CO(n,R). The purpose of this talk is to investigate to what extent the corresponding statement is true for their discrete analogues. It turns out that it does not always hold. |
| Date: | October 4 (Tue), 2005, 16:30-17:30 |
| Room: | RIMS, Kyoto University : Room 402 |
| Speaker: | Troels Johansen (University of Paderborn) |
| Title: | Geometry of Orbits and the Plancherel Decomposition |
| Abstract: [pdf] |
For a class of affine symmetric spaces G/H, we construct a tube domain whose Silov boundary S allows for a G-equivariant identification of the associated L^2-spaces. The adjoint orbits of H in the tangent space of G/H are realized as open orbits in the abelian group S, and we associate natural unitary representations to these orbits. In the rank one case we thus obtain the Plancherel decomposition. Furthermore the orthogonal projection onto 'the most continuous part' may be described by the (un-normalized) orbital integral associated to the Cartan subspace with noncompact centralizer group in H. |
| Date: | September 2 (Fri), 2005, 13:30-14:30 |
| Room: | RIMS Room 402 |
| Speaker: | Alexander Alldridge (University of Paderborn) |
| Title: | The Embedding of Discrete Series Representations of Facial Subgroups |
| Abstract: [pdf] |
Consider a Hermitian symmetric domain B with connected automorphism group G. The boundary of the convex domain B decomposes into lower-rank Hermitian symmetric spaces \bar B with connected automorphism groups \bar G\subset G. It is natural to ask for embeddings of discrete series representations (or more general irreducible unitary representations) of \bar G into corresponding representations of G. If B is an irreducible classical domain, we exhibit an explicit unitary embedding of all discrete series representations of \bar G (holomorphic or non-holomorphic), such that the highest weight vectors of the lowest K-types correspond. The construction uses Knapp-Wallach's Szeg\"o operators, and can be extended to all representations in the support of the Plancherel measure of \bar G. |
| Date: | September 2 (Fri), 2005, 15:00-16:00 |
| Room: | RIMS Room 402 |
| Speaker: | ΝY Takeshi Kawazoe (Keio University) |
| Title: | On Hardy's theorem on SU(1,1) |
| Abstract: [pdf] |
The classical Hardy theorem asserts that $f$ and its Fourier transform $\hat f$ can not be very rapidly decreasing. This theorem was generalized on Lie groups by various people, and also for the Fourier-Jacobi transform. Especially, the heat kernel plays an essential role, which is a "good" function in the sense that $f$ and a generalised Fourier transform both have good decay. However, on SU(1,1) there are infinitely many "good" functions. In this talk, we shall consider a characterization of "good" functions on SU(1,1). |
| Date: | March 18 (Fri), 2005, 18:00-19:00 |
| Room: | RIMS Room 402 |
| Speaker: | Dan Barbasch (Cornell University) |
| Title: | Relevant and petite K-types for real and p-adic groups |
| Abstract: [pdf] |
The unitary dual of a reductive group over a local field plays an important role in noncommutative harmonic analysis. Its structure is also relevant for many problems in analysis, mathematical physics and automorphic forms. In this talk I will survey progress on the determination of the unitary dual. In particular, relevant W-types are a set of representations of the Weyl group which give necessary and sufficient conditions for determining the spherical unitary dual of a split p-adic group. Petite K-types are represntations of the maximal compact subgroup of a real group which are closely related to the relevant W-types. They provide a means to transfer results about the unitary dual of p-adic groups to the case of various series of representations of real groups. |
| Date: | February 24 (Thu), 11:00-12:00 | ||||||||||||||||
| Room: | RIMS Room 005 | ||||||||||||||||
| Speaker: | an Pm Wachi Akihito | ||||||||||||||||
| Affiliation: | Hokkaido Institute of Technology | ||||||||||||||||
| Title: | Capelli identities for symmetric pairs of non-Hermitian type | ||||||||||||||||
| Abstract: [pdf] |
Consider a see-saw pair of real reductive Lie groups
in the real symplectic group $Sp_{2N}(R)$,
Let $\omega$ be the Weil (oscillator) representation of $Sp_{2N}(R)$. Then we have the equality, $\omega( U(g)^K ) = \omega( U(m)^H )$, where $g$ is the complexified Lie algebra of $G_0$, $K$ is the complexification of $K$, and $U(g)^K$ is the set of $K$-invariants of $U(g)$. When $(G_0, K_0)$ is a symmetric pair of Hermitian type, we have already given the Capelli identities, which expresses particular elements of $U(g)^K$ by $U(m)^H$ in the image of $\omega$. In this talk, we give the Capelli identities, which conversely expresses particular elements of $U(m)^H$ by $U(g)^K$ for the see-saw pair called Case C:
|
| Date: | February 23 (Wed), 16:30-17:30 |
| Room: | RIMS Room 202 |
| Speaker: | Michel Duflo (Paris) |
| Title: | Restrictions of discrete series of semisimple Lie groups |
| Abstract: [pdf] |
In this lecture, I will recall first the classification of discrete series representations of real algebraic Lie group, in the setting of the orbit method, and discuss related properties of Lie algebras. In the case of reductive groups, I will present some results on their restrictions to closed subgroup. |
| Date: | January 20 (Thu), 17:00-18:00 |
| Room: | RIMS Room 005 |
| Speaker: | R a¬@iKazunari Sugiyama) |
| Title: | Multiplicity One Property and the Decomposition of b-Functions |
| Abstract: [pdf] |
This talk is based on a joint work with Professor Fumihiro Sato
(Rikkyo University).
Recently, extensive calculations have been made on b-functions of prehomogeneous vector spaces with reducible representations. By examining the results of these calculations, we observe that b-functions of a large number of reducible prehomogeneous vector spaces have decompositions corresponding to those of representations. In this talk, we show that such phenomena can be ascribed to a certain multiplicity one property for group actions on polynomial rings. Our method can be applied equally to non-regular prehomogeneous vector spaces. |