Orbit method

Abstract: This survey is the expanded version of my talk at the AMS meeting in April 1997. I explain to non-experts how to use the orbit method, discuss its strong and weak points and advertise some open problems.

Abstract: 1. The Independent University of Moscow and student sessions at the IUM 2. Mysterious mathematical trinities V. I. Arnold 3. The principle of topological economy in algebraic geometry V. I. Arnold 4. Rational curves, elliptic curves, and the Painleve equation Yu. I. Manin 5. The orbit method and finite groups A. A. Kirillov 6. On the development of the theory of dynamical systems during the past quarter century D. V. Anosov New or 'renewed' directions 'Named' problems Some other achievements 7. Foundations of computational complexity theory A. A Razborov 8. The Schrodinger equation and symplectic geometry S. P. Novikov 9. Rings and algebraic varieties Miles Reid 10. Billiard table as a playground for a mathematician A. B. Katok 11. The Fibonacci numbers and simplicity of 2127 minus 1 A. N. Rudakov 12. On problems of computational complexity Stephen Smale 13. Values of the -function Pierre Cartier 14. Combinatorics of trees Pierre Cartier 15. What is an operad Pierre Cartier? 16. The orbit method beyond Lie groups A. A. Kirillov Infinite-dimensional groups 17. The orbit method beyond Lie groups A. A. Kirillov Quantum groups 18. Conformal mappings and the Ehitham equations I. M. Krichever 19. Projective differential geometry: old and new V. Yu. Ovsienko 20. Haken's method of normal surfaces and its applications to classification problem for 3-dimensional manifolds - the life story of one theorem S. V. Matveev.

Abstract: The general linear group GL(n, K) over a field K contains a particularly prominent subgroup U(n, K), consisting of all the upper triangular unipotent elements. In this paper we are interested in the case when K is the finite field F_q, and our goal is to better understand the representation theory of U(n, F_q). The complete classification of the complex irreducible representations of this group has long been known to be a difficult task. The orbit method of Kirillov, famous for its success when K has characteristic 0, is a natural source of intuition and conjectures, but in our case the relation between coadjoint orbits and complex representations is still a mystery. Here we introduce a natural variant of the orbit method, in which the central role is played by certain clusters of coadjoint orbits. This "method of clusters" leads to the construction of a subring in the representation ring of U(n, F_q) that is rich in structure but pleasantly comprehensible. The cluster method also has many of the major features one would expect from the philosophy of orbit method.