  Partially motivated by the fact that the grand partition function of the ABJM
theory or its generalization is expressed by a spectral operator enjoying
symmetries of the Weyl group, it was found that the grand partition function
satisfies the q-Painleve equation, which is constructed from the affine Weyl
group. In this paper we clarify the affine symmetries of the grand partition
function. With the affine symmetries, we find that the grand partition function
extends naturally outside the fundamental domain of duality cascades and once
the Painleve equation holds in the fundamental domain, so does it outside.
