**Context**


Let $V$ be a 2-dimensional evaluation representation of the quantum loop algebra $\mathcal{U}\_{q}\left(\mathcal{L}\mathfrak{sl}\_{2}\right)$ with $a=q$. Also, for $m\in\mathbb{Z}$, the Drinfeld generator $x\_{m}^{+}$ ($x^{-}\_{m}$) acts by $k^{m}X$ ($k^{m}Y$) on $V$.


We recall that $\mathcal{U}\_{q}\left(\mathcal{L}\mathfrak{sl}\_{2}\right)$ contains the Chevalley generators $X\_{1}=x^{+}\_{0}$, $X\_{0}=k^{-1}x^{+}\_{1}$, $Y\_{1}=x^{-}\_{0}$, $Y\_{0}=x^{-}\_{-1}k$ with the coproduct
$$
\triangle\left(X\_{0}\right)=X\_{0}\otimes1+k^{-1}X\_{0},\triangle\left(X\_{1}\right)=X\_{1}\otimes1+k\otimes X\_{1},\triangle\left(k\right)=k\otimes k
$$
$$
\triangle\left(Y\_{0}\right)=Y\_{0}\otimes k+1\otimes Y\_{0},\triangle\left(Y\_{1}\right)=Y\_{1}\otimes k^{-1}+1\otimes Y\_{1}
$$
and for $m,m^\prime\ge0$ such that $m+m^\prime\ne0$:
$$
\left[x\_{-m}^{+},x\_{-m^{\prime}}^{-}\right]=\phi\_{-m-m^{\prime}}/\left(q^{-1}-q\right),
$$
where
$$
\phi^{-}\left(z\right)=k^{-1}\exp\left(\left(q^{-1}-q\right)\sum\_{m>0}h\_{-m}z^{-m}\right)=\sum\_{m\ge0}\phi\_{-m}z^{-m}
$$


**Problem**


Consider
$$
T\left(z\right)=\exp\left(\sum\_{m>0}\frac{h\_{-m}\left(q-q^{-1}\right)z^{-m}}{\left(q^{2m}-q^{-2m}\right)}\right)
$$


Let $g\left(z\right)$ be the eigenvalue of $T\left(z\right)$ over $\left(V\right)\_{q}$.


How to show that the eigenvalues of $\left(g\left(z\right)\right)^{-1}T\left(z\right)$ on $V\left[\left[z\right]\right]$ are $1$ and $1-zq^{-1}$?


Along with that, I would like to understand better how $T\left(z\right)$ acts on $V\left[\left[z\right]\right]$.


**My thoughts so far**


The $1$ eigenvalue is obvious, with $\left(V\right)\_{q}$ being an eigenspace. I think that $\left(V\right)\_{q}z$ is probably an eigenspace corresponding to eigenvalue $1-zq^{-1}$. However, I'm not sure how to get the result. Perhaps it can be done using Drinfeld polynomials or considering eigenvalues of $h\_{-m}$ and noticing something about $T\left(z\right)$.



How does an element $T\left(z\right)$ act on a $\mathcal{U}_{q}\left(\mathcal{L}\mathfrak{sl}_{2}\right)\left[\left[z\right]\right]$-module?
For the meaning of $T(z)$, see the paper of Frenkel and Hernandez (<http://arxiv.org/abs/1308.3444>, version 4) Example 5.6. The eigenvalues of $T(z)$ are related to the so-called Frenkel-Reshetikhin q-character of $V$. See Proposition 5.8 of that paper for the precise statement and the proof.


