Lecture series by Dmitry Lebedev (3 & 4/6) (ITEP, Moscow)

Title: Lectures on Whittaker function

2014.06.04 | Jane Jamshidi

Date Thu 19 Jun
Time 14:15 16:15
Location Fysisk Auditorium (1523-318)


     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.