On Euler Characteristic of equivariant sheaves

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\ell$ be another prime number. O. Gabber and F. Loeser proved that for any algebraic torus $T$ over $k$ and any perverse $\ell$-adic sheaf \calF on $T$ the Euler characteristic \chi(\calF) is non-negative. We conjecture that the same result holds for any perverse sheaf \calF on a reductive group $G$ over $k$ which is equivariant with respect to the adjoint action. We prove the conjecture when \calF is obtained by Goresky-MacPherson extension from the set of regular semi-simple elements in $G$. From this we deduce that the conjecture holds for $G$ of semi-simple rank 1

Finite difference quantum Toda lattice via equivariant K-theory

We construct the action of the quantum group U_v(sl_n) by the natural correspondences in the equivariant localized $K$-theory of the Laumon based Quasiflags' moduli spaces. The resulting module is the universal Verma module. We construct geometrically the Shapovalov scalar product and the Whittaker vectors. It follows that a certain generating function of the characters of the global sections of the structure sheaves of the Laumon moduli spaces satisfies a $v$-difference analogue of the quantum Toda lattice system, reproving the main theorem of Givental-Lee. The similar constructions are performed for the affine Lie agebra \hat{sl}_n.Comment: Some corrections are made in Sections 2,

Twisted zastava and qq-Whittaker functions

In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces.Comment: 18 pages. v4: the final published versio