  We construct Gaiotto states with fundamental multiplets in $SU(N)$ gauge
theories, in terms of the orthonormal basis of spherical degenerate double
affine Hecke algebra (SH in short), the representations of which are equivalent
to those of $W_n$ algebra with additional $U(1)$ current. The generalized
Whittaker conditions are demonstrated under the action of SH, and further
rewritten in terms of $W_n$ algebra. Our approach not only consists with the
existing literature but also holds for general $SU(N)$ case.
