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

Alexander Braverman, Michael Finkelberg
2014 Journal of The American Mathematical Society  
Theorem 1.5. Assume that g is simply laced. Then the functions J α satisfy the following recursive relation: Here β → β * stands for the natural isomorphism between the coroot lattice of g and its root lattice. The equation (1.3) appears in [14] , where the authors show that (1.3) holds precisely if and only if the generating function of the J α 's is an eigen-function of the above-mentioned quantum difference Toda system. Thus, Theorem 1.5 and the main result of [14] imply the following. 1 In
more » ... act, in [24] the authors work with an arbitrary smooth projective variety X instead of B g . In this case the definition of J α is similar, however technically the pushforward must be taken with respect to certain virtual fundamental cycle in K-theory. In the case when X is a homogeneous space of a linear algebraic group, this reduces to the usual pushforward. License or copyright restrictions may apply to redistribution; see SEMI-INFINITE SCHUBERT VARIETIES AND QUANTUM K-THEORY 1149 Corollary 1.6. Let g be simply laced. Then the equivariant K-theoretic J-function of B g is an eigen-function of the quantum difference Toda integrable system associated with g.
doi:10.1090/s0894-0347-2014-00797-9 fatcat:3xvrovlzyrcxjjznqt5qptndxy