We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory.
An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.
@inproceedings{davidon,
title={On the finiteness of quantum K-theory of a homogeneous space},
author={David Anderson, Linda Chen, and Hsian-Hua Tseng},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190702224334788937395},
}
David Anderson, Linda Chen, and Hsian-Hua Tseng. On the finiteness of quantum K-theory of a homogeneous space. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190702224334788937395.