  We consider the $k$-twisted Nekrasov-Shatashvili limit (NS$_k$ limit) of 5d
(K-theoretic) and 6d (elliptic) quiver gauge theory, where one of the
multiplicative equivariant parameters is taken to be the $k$-th root of unity.
We obtain the extended center of the associated $q$-deformed quiver W-algebras
constructed by our formalism [arXiv:1512.08533; arXiv:1608.04651;
arXiv:1705.04410], which provides gauge theoretic proof of Bouwknegt-Pilch's
statement on the relation to the representation ring of quantum affine algebra.
