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Lectures on Whittaker function (3/6)

Dmitry Lebedev
(ITEP, Moscow and Independent University of Moscow (IUM))
Minikursus
Torsdag, 19 juni, 2014, at 14:15-15:15, (1523-318) Fysisk Auditorium
Abstrakt:
Whittaker function was defined as a matrix element in principal series representation of a reductive Lie group (Jacquet, 1967). Later, it was discovered that in the case of the real reductive group the restriction of the Whittaker function to the Cartan torus (up to a phase multiplier) defines an eigenfunction of an open quantum Toda chain. This allows to put a wide set of methods and tools of solving quantum integrable systems into the framework of representation theory of reductive groups.
In several lectures I will give an account of various group-theoretic constructions and basic properties of the Whittaker function and introduce certain classes of their integral representations. Then I will introduce a tropical limit of the GL(N,R)-Whittaker function and show that the "tropical GL(N)-Whittaker function" coincides with the equivariant symplectic volume of flag manifold of GL(N). Relation with the equivariant version of the BGG (Bernstein, Gelfand, Gelfand) theory will be discussed in details. Finally I will describe a connection of the Whittaker function with Macdonald's theory of symmetric polynomials.
The lectures are based on my recent results obtained with A. Gerasimov and S. Oblezin.
Organiseret af: QGM
Kontaktperson: Jørgen Ellegaard Andersen