On the universal weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$

The investigation is continued of the universal weight function for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_N)$. Two recurrence relations are obtained for the universal weight function with the help of the method of projections. On the level of the evaluation representation of $U_q(\widehat{\mathfrak{gl}}_N)$, two recurrence relations are reproduced, which were calculated earlier for the off-shell Bethe vectors by combinatorial methods. One of the results of the paper is a description of two different but isomorphic currents or “new” realizations of the algebra $U_q(\widehat{\mathfrak{gl}}_N)$, corresponding to two different Gauss decompositions of the fundamental $\mathrm{L}$-operators.