Quantum cohomology of the infinite dimensional generalized flag manifolds

Abstract

Consider the infinite dimensional flag manifold $LK/T$ corresponding to the simple Lie group $K$ of rank $l$ and with maximal torus $T$. We show that, for $K$ of type $A$, $B$ or $C$, if we endow the space $H^*(LK/T)\otimes \bR[q_1,...,q_{l+1}]$ (where $q_1,...,q_{l+1}$ are multiplicative variables) with an $\bR[\{q_j\}]$-bilinear product satisfying some simple properties analogous to the quantum product on $QH^*(K/T)$, then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism type of $QH^*(K/T)$ is determined by the integrals of motion of the non-periodic Toda lattice (see the theorem of Kim). This is a generalization of a theorem of Guest and Otofuji.