## Project Themes

Our inspiration for the project can be viewed in the framework
of the following interrelated themes.

The Orbit Method tackles the fundamental question of how
to describe representations of Lie algebras. The key observation is that
coadjoint orbits admit a symplectic structure akin to phase space of classical
mechanics so that the representations are to be obtained by a procedure
mirroring the passage to quantum mechanics. Here the basic issue is how
to replace one set of the conjugate variables by derivations of the first
set, thereby encorporating the construction leading to the Heisenberg uncertainty
principle. Since its inception during the sixties, arising in the work
of mainly Kirillov, Kostant and Souriau, the orbit method has required
an increasing number of geometric and algebraic techniques to handle some
of the very delicate situations which can arise. In recent developments
the more general notion of a symplectic leaf takes the role of a coadjoint
orbit.

Already since its very beginnings, in the end of the nineteenth
century, representation theory has been seen to contain a very rich and
intricate combinatorial structure. Early manifestations include the theory
of Young tableaux, the Littlewood-Richardson rule and the Weyl denominator
formula. To all this has been added in a quite spectacular development,
the theory of crystal or canonical basis arising in the work of Lusztig,
Kashiwara and Littelmann. This new theory not only extends earlier combinatorial
results; but gives a precise method ( the "crystallization" limit)
to extract the combinatorics from the defining algebraic structure.

Quantum groups developed particularly by Drinfeld to understand
exactly solvable models of statistical physics have added a whole new range
of relationships and problems connected to the theory of semisimple Lie
algebras. The possibility to interpret the underlying parameter in Quantum
Groups as a p-th root of unity has given a new element to the study of
representations of algebraic groups involving modular representations,
Kac-Moody theory and tilting modules.

Important contributions to Number Theory have been made
through representations of algebraic groups, coming to its apogee in the
Langlands programme. In this and in representation theory in general, the
functorial approach has become a powerful calculative tool and a dominant
conceptual framework . This approach has proved particularly powerful in
the formulation of the geometric Langlands conjecture and it also suggests
the existence of a localization theorem for the representations of p-adic
groups. Recent developments have shown how such abstract techniques can
yield solutions to famous "elementary" classical problems.

The study of enveloping algebras and particularly of their
primitive spectra was developed through the inspiration of Dixmier and
Gabriel in the late sixties and early seventies. Notable invariants of
the resulting theory involved nilpotent orbits, Goldie ranks and K-theory
which together gave a remarkable connection to the Springer correspondence
relating orbit closures to Weyl group representations. The same correspondence
has now been seem to arise in a study of differential operators inspired
by the work of Harish-Chandra and underlies one of the many remarkable
connections beween geometry and representation theory which can be so brought
to light.

Gabriele Bogner e-mail:
bogner@sun6.mathematik.uni-freiburg.de

13. Oktober 2000