# Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

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.