A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of the Virasoro algebra) [1].
If the representation is unitary and irreducible, then $c$ and $h$ are non-negative real numbers, and for $c<1$ we have an expression for $h(c)$ ($h$ in terms of $c$) [2].

My question is: is there such an expression $h(c)$ for $c\ge 1$?
(In particular, I am interested in the case of the Grassmannian coset model $$\frac{\widehat{S U}(n+1)_{k} \times \widehat{S O}(2 n)_{1}}{\widehat{S U}(n)_{k+1} \times \widehat{U}(1)_{n(n+1)(k+n+1)}}, \quad n, k \in \mathbb Z^+,$$ which is a coset model and thus a representation of the Virasoro algebra, and has central charge $\frac{3nk}{n+k+1}$.)

Edit: In fact, for $c\ge 1$ we have unitary HW irreps for any pair $(c,h), h\ge 0$. But the question for my particular case remains: for what $h$ the representations $(\frac{3nk}{n+k+1},h)$ label representations come from the Grassmannian model?

[1] Kac, V. G., Raina, A. K., and Rozhkovskaya, N. (2013). Bombay lectures on highest weight representations of infinite dimensional Lie algebras, volume 29. World scientific.

[2] Goddard, P., Kent, A., and Olive, D. (1986). Unitary representations of the Virasoro and super-Virasoro algebras. Communica- tions In Mathematical Physics, 103(1):105–119.


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