On the finiteness of quantum K-theory of a homogeneous space

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.