The Orbit Method in Geometry and Physics In Honor of A.A. Kirillov

The volume is dedicated to AA. Kirillov and emerged from an international con­ ference which was held in Luminy, Marseille, in December 2000, on the occasion 6 of Alexandre Alexandrovitch's 2 th birthday. The conference was devoted to the orbit method in representation theory, an important subject that influenced the de­ velopment of mathematics in the second half of the XXth century. Among the famous names related to this branch of mathematics, the name of AA Kirillov certainly holds a distinguished place, as the inventor and founder of the orbit method. The research articles in this volume are an outgrowth of the Kirillov Fest and they illustrate the most recent achievements in the orbit method and other areas closely related to the scientific interests of AA Kirillov. The orbit method has come to mean a method for obtaining the representations of Lie groups. It was successfully applied by Kirillov to obtain the unitary rep­ resentation theory of nilpotent Lie groups, and at the end of this famous 1962 paper, it was suggested that the method may be applicable to other Lie groups as well. Over the years, the orbit method has helped to link harmonic analysis (the theory of unitary representations of Lie groups) with differential geometry (the symplectic geometry of homogeneous spaces). This theory reinvigorated many classical domains of mathematics, such as representation theory, integrable sys­ tems, complex algebraic geometry. It is now a useful and powerful tool in all of these areas.

A Principle of Variations in Representation Theory.- Finite Group Actions on Poisson Algebras.- Representations of Quantum Tori and G-bundles on Elliptic Curves.- Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models I.- Brèves remarques sur l’oeuvre de A. A. Kirillov.- Gerbes of Chiral Differential Operators. III.- Defining Relations for the Exceptional Lie Superalgebras of Vector Fields.- Schur-Weyl Duality and Representations of Permutation Groups.- Quantization of Hypersurface Orbital Varieties insln.- Generalization of a Theorem of Waldspurger to Nice Representations.- Two More Variations on the Triangular Theme.- The Generalized Cayley Map from an Algebraic Group to its Lie Algebra.- Geometry ofGLn(?)at Infinity: Hinges, Complete Collineations, Projective Compactifications, and Universal Boundary.- Why Would Multiplicities be Log-Concave?.- Point Processes Related to the Infinite Symmetric Group.- Some Toric Manifolds and a Path Integral.- Projective Schur Functions as Bispherical Functions on Certain Homogeneous Superspaces.- Maximal Subalgebras of the Classical Linear Lie Superalgebras.