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We construct Gaiotto states with fundamental multiplets in $SU(N)$ gaugetheories, in terms of the orthonormal basis of spherical degenerate doubleaffine Hecke algebra (SH in short), the representations of which are equivalentto those of $W_n$ algebra with additional $U(1)$ current. The generalizedWhittaker conditions are demonstrated under the action of SH, and furtherrewritten in terms of $W_n$ algebra. Our approach not only consists with theexisting literature but also holds for general $SU(N)$ case.

Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA