Evaluation-type deformed modules over the quantum affine vertex algebras of type A

Let Vᶜ(glN) be Etingof-Kazhdan's quantum affine vertex algebra associated with the trigonometric R-matrix. We establish a connection between suitably generalized deformed $ϕ$-coordinated Vᶜ(glN)-modules and the representations of quantized enveloping algebra Uₕ(glN) and reflection equation algebra Oₕ(MatN). As an application, we demonstrate how the elements of the center of Vᶜ(glN) at the critical level $c=-N$ give rise to the q-analogues of quantum immanants for Uₕ(glN), which were recently found by Jing, Liu and Molev. Finally, we derive the analogues of these results for the quantum affine vertex algebra associated with the normalized Yang R-matrix.