In this [post](https://mathoverflow.net/questions/277803/kazhdan-lusztig-theorem-for-composition-factors-of-verma-modules), Humphreys provides a reference concerning about Kazhdan-Lusztig Conjecture for arbitrary weights in $\mathfrak{h}^\*$, which is under the name: **[Kashiwara and Tanisaki - Characters of irreducible modules with
non-critical highest weights over affine Lie algebras](https://arxiv.org/abs/math/9903123)**.


In other words, for complex semisimple Lie algebra, the problem for expressing $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M(\mu)$ is completely answered, where $M(\eta)$ is the Verma module with weight $\eta$ and $L(\eta)$ is its unique simple quotient.



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> What is about the problem for expressing $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M\_I(\mu)$, where $M\_I(\eta)$ is the parabolic Verma module with weight $\eta$ and $L(\eta)$ is its unique simple quotient. Is it still a Conjecture or is it just easily derived from the expression of $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M(\mu)$?
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As [pointed out](https://mathoverflow.net/a/339250) by Rafael Mrđen, the above problem has been solved for arbitrary regular integral weights. 



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> I would like to know the story for **singular non-integral** weights. Is it the same formula? I believe there should be some difference as in the case for ordinary Kazhdan-Lusztig Conjecture. For example, one need the integral Weyl group of $\lambda$: $W\_{[\lambda]}:=\langle s\_\alpha\in W:\alpha\in \Phi\_{[\lambda]}\rangle$, where $\Phi\_{[\lambda]}:=\{\alpha\in\Phi:\langle\lambda,\alpha^\lor\rangle\in\mathbb{Z}\}$ in order to state the character formula of $L(\lambda)$ for non-integral $\lambda$. I also believe that singularity may play a role in the formula.
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> I hope to see the explicit formula for $\mathrm{ch}L(\lambda)$ in terms of $\mathrm{ch}M\_I(\mu)$ and some kinds of parabolic KL polynomials, where $\lambda$ is singular non-integral. Any paper talks about this explicitly?
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Parabolic Kazhdan-Lusztig Conjecture
See section 9.7. (Relative Kazhdan-Lusztig Theory) of his book [1], and the references therein.




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[1] J. Humphreys. [Representations of semisimple Lie algebras in the BGG category $\mathscr O$](http://www.ams.org/books/gsm/094). Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008.


