After reviewing the elliptic quantum group $U_{q,p}(g)$ associated with the affine Lie algebra $g$, we introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. By using the vertex operators (the intertwining operators) of the $U_{q,t,p}(gl_{1,tor})$-modules w.r.t. the Drinfeld comultiplication, we give a realization of the affine quiver W-algebra $W_{q,t}(\Gamma(\widehat{A_0}))$ proposed by Kimura–Pestun. This realization turns out to be useful to derive the Nekrasov instanton partition functions of the 5d and 6d lifts of the 4d $N = 2^∗$ theories, i.e. the generating functions of the $\chi_y$- and the elliptic genus of the instanton moduli spaces, and provide a new Alday–Gaiotto–Tachikawa correspondence.
