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

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.

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.