Let $a \in \mathbb{C}^{\times}$, $r \in N$. Let $W = V\_q(r)$ be the $r$-dimensional irreducible type 1 representation of $U\_q(gl\_2(\mathbb{C}))$. In the usual basis $\{v\_0, \ldots, v\_r\}$, the action of $U\_q(gl\_2(\mathbb{C}))$ on $V\_q(r)$ is given by 
\begin{align}
& t\_0.v\_p = q^{(r-2p)/2} v\_p, \\
& t\_1.v\_p = q^{-(r-2p)/2} v\_p, \\
& x\_1^+.v\_p = [r-p+1]\_q v\_{p-1}, \\
& x\_1^{-}.v\_p = [p+1]\_q v\_{p+1}.
\end{align}


The evaluation module $V\_q(r)\_a$ is the pull-back of the module $
V\_q(r)$ via the evaluation map $ev\_a: U\_q(L(sl\_2)) \to U\_q(gl\_2)$.


On page 401, Example 12.2.11, of the book [a guide to quantum groups](https://books.google.co.il/books?id=bn5GNkLfnsAC&redir_esc=y), it is said that the Drinfeld polynomial associated to the evaluation module $V\_q(r)\_a$ is 
$$
P\_1(u) = \prod\_{p=1}^r(u - a^{-1} q^{2p-r-1}).
$$
How to compute this polynomial? Thank you very much.



Evaluation modules of $U_q(L(sl_2))$
The answer of this problem is given in [the paper](https://projecteuclid.org/euclid.cmp/1104248585) by Chari and Pressley (the Corollary on Page 272). 


