Bounded symmetric domains, their compact duals and related symmetric spaces are a rich ground for beautiful mathematical theories, whose special flavor consists in an interplay of classical results and modern approaches in algebra, geometry, representation theory and analysis.
This conference, which is the second Luminy conference devoted to these topics (the first has been organised in october 1996 by Jean-Louis Clerc, Jacques Faraut and Hubert Rubenthaler), will focus on the following topics:
It is by now classical that certain symmetric spaces, most notably bounded symmetric domains and symmetric cones, correspond to Jordan algebraic structures, and that Jordan theoretic methods work very well (and better than Lie theoretic methods) in the infinite dimensional case. Therefore geometric aspects of such spaces are closely related (or even equivalent) to Jordan algebraic questions. It seems nowadays that Jordan theoretic objects play a key role in the general theory of infinite dimensional Lie algebras and Lie groups.
Ever since Harish Chandra's famous construction of the holomorphic discrete series, bounded symmetric domains, their compact duals and their various real forms have been a preferred framework for the geometric construction and detailed study of Lie group representations. Since such representations often live in sections of bundles over these spaces, their study is closely related to geometric invariants which, in turn, often generalise classical invariants such as the cross-ratio or the Maslov-index. Research in this area has been very active during the last years, last but no least by the work of Jean-Louis Clerc.
In the general framework of harmonic analysis on symmetric spaces, the special nature of the spaces considered here opens the way for more specific approaches, involving, for instance, complex or CR geometry, causal structures or prehomogeneous vector spaces. Some of the geometric invariants mentioned above show up remarkably often in harmonic analysis, for instance in approaches to quantization and as kernel functions associated to representations.