Evaluation modules for quantum toroidal ${mathfrak{gl}}_n$ algebras

The affine evaluation map is a surjective homomorphism from the quantum toroidal ${mathfrak {gl}}_n$ algebra ${mathcal E}_n(q_1,q_2,q_3)$ to the quantum affine algebra $U_qwidehat{mathfrak {gl}}_n$ at level $kappa$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=kappa^2$. We discuss ${mathcal E}_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${mathcal E}_n(q_1,q_2,q_3)$, which describes a deformation of the coset theory $widehat{mathfrak {gl}}_n/widehat{mathfrak {gl}}_{n-1}$.