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

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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.

This post imported from StackExchange MathOverflow at 2019-08-21 22:28 (UTC), posted by SE-user Soap
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