Project Themes

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

The Orbit Method

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.

Crystals, Canonical Bases

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

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.

Geometric Langlands Conjecture

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.

Differential Operators and Enveloping Algebras

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:
13. Oktober 2000