I am a physicist studying conformal field theory (CFT), so what I state can be not precise.
In physics literature, the affine Lie algebra $\mathfrak{su}(2)_k$ (here $k$ is the level) has only finitely many integrable representation, which can be labelled as $0, 1/2, \ldots, k$. I want to verify this statement by the general mathematical theory of the integrable representation.
The following is what I know (or heard) about the general theory of integrable representation:
Let $\mathfrak g$ be a simple Lie algebra. Then, we define the affine Lie algebra $\tilde{\mathfrak g} = \mathfrak g \otimes \mathbb C(t) \oplus \mathbb Cc \oplus \mathbb Cd$, where $\mathbb C(t)$ is an algebra of Laurent polynomials, $c$ is as an element comes from the central extension, and $d$ can be though of the derivation. We have a triangular decomposition $$\tilde{\mathfrak g} = \tilde{\mathfrak n}_+ \oplus \tilde{\mathfrak h} \oplus \tilde{\mathfrak n}_-.$$
Let $\alpha_1, \ldots, \alpha_r$ be the simple roots of $\mathfrak g$. Then, the simple roots of $\tilde{\mathfrak g}$ is $\alpha_1, \ldots, \alpha_r$ together with $\alpha_0= \delta-\theta$, where $\delta\in \mathfrak{\tilde h}^*$ is defined as $$\delta(d) = 1, \quad \delta(c)=0, \quad \delta|_{\mathfrak h}=0,$$ and $\theta$ is the maximal root of $\mathfrak g$.
It is known that every integrable module is given by the irreducible quotient $L(\tilde\lambda)$ of the Verma module, where the highest weight is $\tilde\lambda$ is a dominant integral weight.
Now we specialize to $\mathfrak g = \mathfrak{su}(2)$ . We know that $\mathfrak g$ has only one simple root $\alpha_1$. Since we consider the level $k$ representation, $c$ always acts as a constant $k$. From 3., we need to find the condition when $\tilde\lambda$ is a dominant integral weight, which is equivalent to check $$\langle \tilde\lambda, \alpha_0\rangle, \langle \tilde\lambda, \alpha_1\rangle \in \mathbb Z_{\geq 0}.$$ This is where I am stuck. In particular, how can I evaluate $\langle \tilde\lambda, \alpha_0\rangle$? Before asking this question, what is a coroot of $\alpha_0$?
