  Brane configurations in a circle allow subsequent applications of the
Hanany-Witten transitions, which are known as duality cascades. By studying the
process of duality cascades corresponding to quantum curves with symmetries of
Weyl groups, we find a hidden structure of affine Weyl groups. Namely, the
fundamental domain of duality cascades consisting of all the final destinations
is characterized by the affine Weyl chamber and the duality cascades are
realized as translations of the affine Weyl group, where the overall rank in
the brane configuration associates to the grading operator of the affine
algebra. The structure of the affine Weyl group guarantees the finiteness of
the processes and the uniqueness of the endpoint of the duality cascades. In
addition to the original duality cascades, we can generalize to the cases with
Fayet-Iliopoulos parameters. There we can utilize the Weyl group to analyze the
fundamental domain similarly and find that the fundamental domain continues to
be the affine Weyl chamber. We further interpret the Weyl group we impose as a
"half" of the Hanany-Witten transition.
