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Article

Semi-infinite Schubert varieties and quantum K-theory of flag manifolds

Journal of the American Mathematical Society. 2014. Vol. 27. No. 4. P. 1147-1168.
Braverman A., Finkelberg M. V.

Let g be a semisimple Lie algebra over complex numbers, and let B be its flag variety. In this paper we study the spaces Z^a of based quasimaps from the projective line to B as well as their affine versions (corresponding to g being untwisted affine algebra). The purpose of this paper is two-fold. First we study the singularities of the above spaces, supposed to model singularities of the not rigorously defined "semi-infinite Schubert varieties". We show that Z^a is normal and when g is simply-laced, Z^a 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^a. When 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=sl(N); in view of the first part this also proves a conjecture by Givental-Lee describing the quantum K-theory of B in terms of the Langlands dual quantum group (for non-simply laced g certain modification of that conjecture is necessary). Similar analysis (modulo certain assumptions) is performed for affine g.