Casimir Operator of $\mathfrak{sl}_n(\mathbb{C})$.

Question

If I have the Lie algebra $\mathfrak{g} = \mathfrak{sl}_n(\mathbb{C})$ and the trace form $B(x,y) = \text{tr}(x,y)$ for $x, y \in \mathfrak{g}$ how does one calculate by which scalar the Casimir element $C \in Z(\mathscr{U}(\mathfrak{g}))$ acts on the highest weight module $V(\lambda)$?

I know that one defines the Casimir by $C = \sum_i x_i x_i^*$ where $\{x_i\}$ is a basis for the Lie algebra and $\{x_i^*\}$ a dual basis, which are arbitrary (i.e. C is independent of the choice of bases).

I have calculated by hand some simple examples (n=3 acts by the scalar $\frac{8}{3}$ for example on V(2)). How should one proceed in general?

Thanks.

Answer 1

The way to calculate the action of $C$ is to make a smart choice of dual bases and then observe that it suffices to check the action of $C$ on a highest weight vector. (The fact that you use the trace form rather than the Killing form shows up only in an overall scaling of $C$.) Basically, you have to observe that the Cartan subalgebra $\mathfrak h$ is orthogonal to all root spaces with respect to $B$ while for two roots $\alpha$ and $\beta$, the restriction of $B$ to $\mathfrak g_{\alpha}\times\mathfrak g_{\beta}$ is non-zero if and only if $\beta=-\alpha$. First choose a basis $\{H_i\}$ of the space $\mathfrak h$ of diagonal matrices which is orthonormal with respect to $B$. Next, for $i