I am trying to solve the following problem, which is related to relatively recent results, but I am not sure how to do it.

**Problem**

In this problem, $\mathfrak{g}=\mathfrak{sl}_{2}$. We study finite-dimensional representations of loop algebra $L\mathfrak{g}$. Let $a\in\mathbb{C}^{*}$ and $\tilde{\lambda}_{a}\in\left(L\mathfrak{h}\right)^{*}$ defined by $\tilde{\lambda}_{a}\left(t^{r}H\right)=a^{r}$ for all $r\in\mathbb{Z}$.

- Show that the representation $L\left(\tilde{\lambda}_{a}\right)$ of the
**$l$-highest weight**is finite-dimensional [one can use the**morphism of evaluation**$ev_{a}:t^{t}\otimes x\rightarrow a^{r}x$]. Specify its dimension and its**$l$-character**. - Show that the module $P_{a}=L\left(\tilde{\lambda}_{a}\right)\otimes L\left(\tilde{\lambda}_{a}\right)$ is not simple.
- Is $P_{a}$ semi-simple?

**Some results of my attempts to solve it**

I am not sure what are the definitions of $l$-weights and $l$-characters of $L\mathfrak{g}$, but I intuitively applied the formulas $\left[t^{r}\otimes x,t^{k}\otimes y\right]=t^{r+k}\left[x,y\right]$ and $\left[E^{\pm},H\right]=\pm2H$ to the assumed expression for highest $l$-weight and highest $l$-weight vector $\left(t^{r}\otimes H\right)\left(\Lambda\right)=a^{r}\Lambda$.

From the calculations, it appears that the "weights" are $a^{r}$, $a^{r}-2t^{r}$, $a^{r}-4t^{r}$, $\cdots$, but this does not look right, because I do not expect $t$ to be part of the expressions for $l$-weights (perhaps if an expression for $l$-weight contains $t$, it is not even a weight, but I have never dealt with $l$-weights before, so maybe I'm wrong).

**References**

[1] Chari, Vyjayanthi, *Integrable representations of affine Lie-algebras*,
Invent. Math. 85 (1986), no. 2,317–335.

[2] Chari, Vyjayanthi and Hernandez, David, *Beyond Kirillov-Reshetikhin modules*,
49–81, Contemp. Math., 506, Amer. Math. Soc., Providence, RI, 2010.

[3] Frenkel, Edward, Reshetikhin, Nicolai,
*The q-characters of representations of quantum affine algebras and deformations of W-algebras*, Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999.