FINITE DIFFERENCE QUANTUM TODA LATTICE, AND QUANTUM K-THEORY OF FLAG
MANIFOLDS
by Michael FINKELBERG (Moscow Independend University and University of Chicago)
Abstract:
The eigenfunctions of the quantum Toda lattice differential operators can
be constructed as the Shapovalov scalar product of Whittaker vectors in
the universal Verma module over the Lie algebra sl_n (Kazhdan, Kostant).
There is a similar construction for the quantum group U_q(sl_n)
(Sevostyanov). It can be realized geometrically in the equivariant
K-theory of moduli spaces of (quasi)maps from the projective line to the
flag variety of sl_n. If follows that a certain generating function of
characters of rings of global functions on these moduli spaces satisfies
the q-difference quantum Toda lattice equation. This function governs
the quantum K-theory of the flag variety (Givental, Lee).