## Abstract

Let g be a semi-simple Lie algebra over c and let B_{g} be its flag variety. In this paper we study the spaces Z_{a}^{g} of based quasi-maps P^{1}→B_{g} (introduced by Finkelberg and Mirković in 1999) as well as their affine versions (corresponding to 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_{a}^{g}is normal and when g is simply laced, Z_{a}^{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_{a}^{g}. 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=S[(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-theory of B_{g} in terms of the Langlands dual quantum group U_{q}(ǧ) (for non-simply laced g certain modification of that conjecture is necessary). Similar analysis (modulo certain assumptions) is performed for affine g, extending the results of Braverman and Finkelberg.

Original language | English |
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Pages (from-to) | 1147-1168 |

Number of pages | 22 |

Journal | Journal of the American Mathematical Society |

Volume | 27 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Oct 2014 |

Externally published | Yes |