This question may be seen as a follow up of this one.

Let $\mathfrak{g}$ be a rank $r$ simple complex Lie algebra, and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. We call $\alpha_i$ the simple roots and $\varpi_i$ the fundamental weights (for $i=1,\dots ,r$). Define the set of integral weights $$\Lambda = \mathbb{Z} \varpi_1 + \dots + \mathbb{Z} \varpi_r$$ and the set of dominant integral weights $$\Lambda^+ = \mathbb{Z}^+ \varpi_1 + \dots + \mathbb{Z}^+ \varpi_r \, . $$

*[EDIT: I originally defined here a partial ordering on the weight which was unfortunate and non necessary for my question. ]*

Introduce finally the Weyl vector $\rho = \varpi_1 + \dots + \varpi_r $, the Weyl group $W$, and the dot action $w \cdot \lambda = w(\lambda + \rho) - \rho$ for $w \in W$. For an integral weight $\lambda$, we call $M(\lambda)$ the associated Verma module and $L(\lambda)$ the associated simple module.

Let $\lambda \in \Lambda$. If $\lambda \in -2\rho - \Lambda^+$, then the Kazhdan-Lusztig Theorem tells us that for any $w \in W$, $$\mathrm{ch}\, L (w \cdot \lambda) = \sum\limits_{x \leq w} (-1)^{\ell (x,w)} P_{x,w} (1) \, \mathrm{ch}\, M (x \cdot\lambda) \, , $$ where we use the Bruhat ordering and the length function on $W$, and the $P_{x,w} (q)$ are the Kazhdan-Lusztig polynomials.

My question is the following. **Let $w \in W$ and $\lambda \in -\rho -\Lambda^+$ such that $\lambda \notin -2 \rho - \Lambda^+$. In other words, there is no element of $\Lambda^+$ in the dot orbit of $\lambda$. Is there a known expression of $\mathrm{ch}\, L (w \cdot \lambda)$ in terms of $\mathrm{ch}\, M (x \cdot\lambda)$, for $x \in W$ ?**