GKO (or coset) construction - all possible highest weights $h$

Question

I am reading the famous paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive.

From a compact simple Lie algebra $\mathfrak{g}$ and a Lie subalgebra $\mathfrak{h}$, they obtain a representation $Vir(\mathfrak{g},\mathfrak{h})$ of the Virasoro algebra. The unitary highest weight irreducible representations of a Virasoro algebra are labelled by $(c,h)$, with $c$ the central charge and $h$ the highest weight. In the paper, they show that $c$ can take any value in the series

In a second moment, they show that $h$ can take any value in the series

They prove this last result using character theory. But what I do not understand is the idea behind this last proof. They start it with the following paragraph:

In particular, I do not understand how to make sense of the highlighted sentence: what do they mean with "decompose with respect to" in this context; how such decomposition helps us at all; and how exactly does (2.20) come to be.

Answer 1

The terminology is explained earlier on that page and the previous page in the paper.

On the same page, we see that they set $\mathfrak{g} = \mathfrak{su}(2) \times \mathfrak{su}(2)$, and let the subalgebra $\mathfrak{h}$ be the diagonal copy of $\mathfrak{su}(2)$. Then, the tensor product of a level $N$ representation of the first $\widehat{\mathfrak{su}(2)}$ and a level 1 representation of the second $\widehat{\mathfrak{su}(2)}$ is called a level $(N,1)$ representation of $\hat{\mathfrak{g}}$. You have to be careful here, because a very similar notation is used to parametrize representations of a single $\widehat{\mathfrak{su}(2)}$: $(N,\ell)$ is the pair where $N$ is the level and $\ell$ is spin.

They pointed out in the previous page that for any embedding $\mathfrak{h} \subset \mathfrak{g}$ of semisimple Lie algebras, the Sugawara construction yields commuting actions of $\hat{\mathfrak{h}}$ and Virasoro on any unitary highest-weight representation of $\hat{\mathfrak{g}}$. Restricting the action of $\hat{\mathfrak{g}}$ to an action of this product yields a decomposition into a sum of tensor products of unitary representations of $\hat{\mathfrak{h}}$ and Virasoro.

This decomposition helps us, because generating unitary representations of Virasoro with all possible parameters $(c,h_{p,q})$ was the goal of the paper, and this construction yields all of them. Equation (2.20) arises as a result of recursively removing submodules generated by highest-weight vectors, using known character formulas to show that the next highest weight is exactly what the equation predicts.