| Language |
English
|
| From |
2009/01/21 16:30
|
| To |
2009/01/21 17:30
|
| Place | 127 Conference Room, Building No.3, Faculty of Science, Kyoto University |
| Seminar Name | GCOE Seminar |
| Title |
Spectral Decomposition for reductive symmetric spaces |
| Field |
Geometry
|
| Speakers | Prof. Patrick Delorme |
| Affiliation | Marseille-Luminy |
| Abstract | The notion of abstract spectral decomposition, or Fourier inversion formula, for an homogeneous space of a Lie group will be described, starting with finite groups. It expresses some class of functions as sums, or series and integrals of ''elementary waves''.
Real reductive symmetric spaces will be defined and some important examples presented, like the group case.
One goal of the lecture is to explicit the abstract Fourier inversion formula for the homogeneous spaces, explaining various aspects and contributions.
The group case has been solved by Harish-Chandra in the 70's. In general the discrete spectrum has been elucidated by Oshima-Matsuki using Flensted-Jensen duality and Poisson transform in the 80's. The contribution of the continuous spectrum has been determined in the 90's by van den Ban-Schlichtkrull and Carmona-D.
The structure of the proof I will present has a lot in common with the spectral decomposition of Langlands for automorphic forms.
In particular one proves a meromorphic continuation of a family of functions: the Eisenstein integrals which are the elementary waves. One studies their asymptotic behavior : theory of the constant term. A process of truncation is also used.
The corresponding problem for p-adic reductive symmetric spaces is still open, although some partial results are available.
A word will be said for Whittaker functions for real and p-adic groups. |
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