As a follow up question of this post.
Let $M_I(\eta)$ be the parabolic Verma module with weight $\eta$ and $L(\eta)$ is its unique simple quotient.
For $\lambda,\mu\in\Lambda_I^+$, can we express $[M_I(\lambda):L(\mu)]$ in terms of Kazhdan-Lusztig polynomials? If this can be done, I hope to see the explicit formula.
P.S. $\lambda$ can be singular or non-integral.
On p.193 of Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$.
What is the current state of the question: When is $[M_I(\lambda):L(\mu)]\neq 0$? Is it still open or anyone knows the answer but no one bother to write it out?
