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, 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.
