  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.
