Let G denote a complex, semisimple, simply-connected group. We identify the equivariant quantum differential equation for the cotangent bundle to the flag variety of G with the affine Knizhnik-Zamolodchikov connection of Cherednik and Matsuo. This recovers Kim's description of quantum cohomology of the flag variety itself as a limiting case. A parallel result is proven for resolutions of the Slodowy slices. Extension to arbitrary symplectic resolutions is discussed.
It is a well-known phenomenon that a particular solution of the quantum differential equation (the so-called @math -function) may often be computed as the generating function of integrals of certain cohomology classes over different compactifications of the moduli space of maps @math . For example for maps to flag varies of @math , one can try to use the so called Laumon moduli space flags of sheaves on @math . The corresponding generating function was identified by A. Negut with the eigenfunctions of the quantum Calogero-Moser system @cite_12 .
@cite_12
2004255580
This paper contains a proof of a conjecture of Braverman concerning Laumon quasiflag spaces. We consider the generating function Z(m), whose coefficients are the integrals of the equivariant Chern polynomial (with variable m) of the tangent bundles of the Laumon spaces. We prove Braverman’s conjecture, which states that Z(m) coincides with the eigenfunction of the Calogero-Sutherland hamiltonian, up to a simple factor which we specify. This conjecture was inspired by the work of Nekrasov in the affine ( sl _ n ) setting, where a similar conjecture is still open.