# 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) .
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 $$c$$) .

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?

 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.

 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.