Every highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by a pair $(c,h)$ of complex numbers [1]. This module can be written as quotient of the unique (up to iso) Verma module with highest weight $(c,h)$.
Since a coset model $G/H$ is a highest weight irreducible Vir-module, then it can be uniquely labelled by such a pair $(c,h)$.
On the other hand, a coset model $G/H$ can also be labelled (non-uniquely) by $(\Lambda,\lambda)$, where $\Lambda$ and $\lambda$ are highest weights of $\mathfrak g$ and $\mathfrak h$, respectively [2].
The Question:
Is there a way to write $h$ and $c$ in terms of $\Lambda$ and $\lambda$? What about a way to write $\Lambda$ and $\lambda$ in terms of $h$ and $c$?
Finally, I know that $h$ can be written in terms of the central charge $c$ for the case $c<1$ [3]. Is it possible to write $h$ in terms of $c$ for when $c \ge 1$?
(I am particularly interested in the coset models $\frac{\widehat{S U}(n+1)_{k} \times \widehat{S O}(2 n)_{1}}{\widehat{S U}(n)_{k+1} \times \widehat{U}(1)_{n(n+1)(k+n+1)}}$, and even more specifically in the case $n=2$).
[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] Nozaki, M. (2002). Comments on d-branes in kazama-suzuki models and landau-ginzburg theories. Journal of High Energy Physics, 2002(03):027.
[3] 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.
