We know for type $A$, there is an evaluation homomorphism from quantum affine algebra to quantum algebra, $$\operatorname{ev}_a:\mathbf{U}_q(\mathbf{L}\mathfrak{g})\to \mathbf{U}_q(\mathfrak{g})$$ for any $a\in \mathbb{C}^\times$ (assume $q$ is not a root of unity). For a simple module $L(\lambda)$ with highest weight $\lambda$ (of type $\mathbf{1}$), the $\operatorname{ev}^*L(\lambda)$ forms a simple module of $\mathbf{U}_q(\mathbf{L}\mathfrak{g})$. My question is what is the Drinfel'd polynomial of it?
In the case of $\mathfrak{g}=\mathfrak{sl}_2$, it is well-known that the Drinfel'd polynomial of $\operatorname{ev}_a^*L(d\omega)$ is given by $$Y_{q^{-d}/a} \cdots Y_{q^{d}/a},\qquad Y_{a}(u)=u-a.$$ But what is the general case?
Does it only involves $Y_{i,x}$ with $x\in q_i^{\mathbb{Z}}/a$?
For example, can we claim the Kirillov--Reshetikhin modules take the same expression?
Can these Drinfel'd polynomials be read from Nakajima quiver varieties (cotangent bundles of partial flag varieties)?
