  In this paper, we prove a class of nontrivial q-Pochhammer symbol identities
with extra parameters by iterated residue method. Then we use these identities
to find relations of the quasi-map $K$-theoretical $I$-functions with level
structure between Grassmannian and its dual Grassmannian. Here we find an
interval of levels within which two $I$-functions are the same, and on the
boundary of that interval, two $I$-functions are intertwining with each other.
We call this phenomenon level correspondence in Grassmann duality.
