The orbit method in geometry and physics in honor of A.A. Kirillov ; [emerged from an international conference ... Luminy, Marseille, in December 2000

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. The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and continues to be an important tool today. Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits in 1962, places him as the founder of orbit theory. The origins of the orbit method lie in the search for a relationship between classical and quantum mechanics. Over the years, the orbit method has been used to link harmonic analysis (theory of unitary representations of Lie groups) with differential geometry (symplectic geometry of homogeneous spaces), and it has stimulated and invigorated many classical domains of mathematics, i.e., representation theory, integrable systems, complex algebraic geometry, to name several. It continues to be a useful and powerful tool in all of these areas of mathematics and mathematical physics.This volume, dedicated to A. A. Kirillov, covers a very broad range of key topics such as: * The orbit method in the theory of unitary representations of Lie groups* Infinite-dimensional Lie groups: their orbits and representations* Quantization and the orbit method; geometric quantization (old and new)* The Virasoro algebra; string and conformal field theories* Lie superalgebras and their representations* Combinatorial aspects of representation theory.The prominent contributors to this volume present original and expository invited papers in the areas of Lie theory, geometry, algebra, and mathematical physics. The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses.Contributors include: A. Alekseev, J. Alev, R. Brylinski, J. Dixmier, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina. Front Matter....Pages i-xiii A Principle of Variations in Representation Theory....Pages 1-7 Finite Group Actions on Poisson Algebras....Pages 9-28 Representations of Quantum Tori and G -bundles on Elliptic Curves....Pages 29-48 Dixmier Algebras for Classical Complex Nilpotent Orbits via Kraft-Procesi Models I....Pages 49-67 Brèves remarques sur l’œuvre de A. A. Kirillov....Pages 69-71 Gerbes of Chiral Differential Operators. III....Pages 73-100 Defining Relations for the Exceptional Lie Superalgebras of Vector Fields....Pages 101-146 Schur-Weyl Duality and Representations of Permutation Groups....Pages 147-164 Quantization of Hypersurface Orbital Varieties in sl n ....Pages 165-196 Generalization of a Theorem of Waldspurger to Nice Representations....Pages 197-242 Two More Variations on the Triangular Theme....Pages 243-258 The Generalized Cayley Map from an Algebraic Group to its Lie Algebra....Pages 259-296 Geometry of GL n (C) at Infinity: Hinges, Complete Collineations,Projective Compactifications, and Universal Boundary....Pages 297-327 Why Would Multiplicities be Log-Concave?....Pages 329-347 Point Processes Related to the Infinite Symmetric Group....Pages 349-393 Some Toric Manifolds and a Path Integral....Pages 395-419 Projective Schur Functions as Bispherical Functions on Certain Homogeneous Superspaces....Pages 421-443 Maximal Subalgebras of the Classical Linear Lie Superalgebras....Pages 445-472 Back Matter....Pages 473-474 The orbit method influenced the development of several areas of mathematics in the second half of the 20th century and remains a useful and powerful tool in such areas as Lie theory, representation theory, integrable systems, complex geometry, and mathematical physics. Among the distinguished names associated with the orbit method is that of A.A. Kirillov, whose pioneering paper on nilpotent orbits (1962), places him as the founder of orbit theory. The original research papers in this volume are written by prominent mathematicians and reflect recent achievements in orbit theory and other closely related areas such as harmonic analysis, classical representation theory, Lie superalgebras, Poisson geometry, and quantization. Contributors: A. Alekseev, J. Alev, V. Baranovksy, R. Brylinski, J. Dixmier, S. Evens, D.R. Farkas, V. Ginzburg, V. Gorbounov, P. Grozman, E. Gutkin, A. Joseph, D. Kazhdan, A.A. Kirillov, B. Kostant, D. Leites, F. Malikov, A. Melnikov, P.W. Michor, Y.A. Neretin, A. Okounkov, G. Olshanski, F. Petrov, A. Polishchuk, W. Rossmann, A. Sergeev, V. Schechtman, I. Shchepochkina. The work will be an invaluable reference for researchers in the above mentioned fields, as well as a useful text for graduate seminars and courses. The principle of variations is one of the basic tools in the theory of partial differential equations, with numerous applications in physics as well, notably as the least action principle in mechanics.