Alexander Braverman

Let G be a semi-simple simply connected group over complex numbers. In this paper we give a geometric definition of the (dual) Weyl modules over the group G[t] and show that their characters form an eigen-function of the lattice version of the q-Toda integrable integrable system (defined by means of the quantum group version of Kostant-Whittaker reduction due to Etingof and Sevostyanov). All the proofs are algebro-geometric and rely on our previous work which interprets the universal eigen-function of the q-Toda system in terms of rings of functions on the spaces of based quasi-maps from P^1 to the flag variety of G. We discuss in detail the relation between the current work and the works of Cherednik, Ion, Sanderson and Gerasimov-Lebedev-Oblezin.

Let g be a semi-simple Lie algebra. In this paper we study the spaces of based quasi-maps from the projective line P^1 to the flag variety of g (it is well-known that their singularities are supposed to model the singularities of the so called semi-infinite Schubert varieties which are hard to define directly). In the first part of the paper we show that the above spaces are normal and in the case when g is simply laced they are also Gorenstein and have rational singularities. In the second part of the paper we compute the character of the ring of functions on the above spaces; in view of the above results this computation can be thought of as a computation of the (equivariant) K-theoretic J-function of the flag variety of g. We show that when g is simply laced the above characters satisfy the "fermionic recursion" version of the difference quantum Toda lattice (due to B.Feigin, E.Feigin, M.Jimbo, T.Miwa and E.Mukhin). As a byproduct we show that the equivariant K-theoretic J-function of the flag variety of a simply laced Lie algebra g is the universal eigen-function of the difference quantum Toda lattice, thus proving a conjecture of Givental and Lee. Some modification of this result is also shown to hold for non-simply laced g. We also discuss an extension of the above results to the case when g is an affine Lie algebra (this extension is conjectural except when g=sl(N)).

Let g \mathfrak {g} be a semi-simple Lie algebra over C \mathbb {C} and let B g \mathcal {B}_{\mathfrak {g}} be its flag variety. In this paper we study the spaces Z g α Z^{\alpha }_{\mathfrak {g}} of based quasi-maps P 1 → B g \mathbb {P}^1\to \mathcal {B}_{\mathfrak {g}} (introduced by Finkelberg and Mirković in 1999) as well as their affine versions (corresponding to g \mathfrak {g} being untwisted affine algebra) introduced by Braverman et al. in 2006. The purpose of this paper is two-fold. First we study the singularities of the above spaces (as was explained by Finkelberg and Mirković in 1999 and Braverman in 2006 they are supposed to model singularities of the not rigorously defined “semi-infinite Schubert varieties”). We show that Z g α Z^{\alpha }_{\mathfrak {g}} is normal and when g \mathfrak {g} is simply laced, Z g α Z^{\alpha }_{\mathfrak {g}} is Gorenstein and has rational singularities; some weaker results are proved also in the affine case. The second purpose is to study the character of the ring of functions on Z g α Z^{\alpha }_{\mathfrak {g}} . When g \mathfrak {g} is finite-dimensional and simply laced we show that the generating function of these characters satisfies the “fermionic formula” version of quantum difference Toda equation, thus extending the results for g = s l ( N ) \mathfrak {g}=\mathfrak {sl}(N) from Givental and Lee in 2003 and Braverman and Finkelberg in 2005; in view of the first part this also proves a conjecture from Givental and Lee in 2003 describing the quantum K K -theory of B g \mathcal {B}_{\mathfrak {g}} in terms of the Langlands dual quantum group U q ( g ˇ ) U_q(\mathfrak {\check {g}}) (for non-simply laced g \mathfrak {g} certain modification of that conjecture is necessary). Similar analysis (modulo certain assumptions) is performed for affine g \mathfrak {g} , extending the results of Braverman and Finkelberg.

This is the third paper of a series (started by arXiv:0711.2083, arXiv:0908.3390) which describes a conjectural analog of the affine Grassmannian for affine Kac-Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian and affine Zastava.

Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the corresponding adelic group. The purpose of this paper is to give a survey of some recent constructions and results, which show that there should exist an analog of the above theories in the case when G is replaced by the corresponding affine Kac-Moody group (which is essentially built from the formal loop group G((t)) of G). Specifically we discuss the following topics : affine (classical and geometric) Satake isomorphism, affine Iwahori-Hecke algebra, affine Eisenstein series and Tamagawa measure.

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a "finite analog" of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from P^1 to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra U(g,e) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of arXiv:math/0401409 when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of certain shifted Yangians.