Parabolic Kazhdan-Lusztig Conjecture

In this post, 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.

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.

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 $$\mathrm{ch}M(\mu)$$?

As pointed out by Rafael Mrđen, the above problem has been solved for arbitrary regular integral weights.

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.

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?

1 Answer

See section 9.7. (Relative Kazhdan-Lusztig Theory) of his book [1], and the references therein.

[1] J. Humphreys. Representations of semisimple Lie algebras in the BGG category $$\mathscr O$$. Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008.

• What is the story for singular non-integral weights? Is it the same formula? – James Cheung Aug 27 '19 at 7:38
• Take a look at Irving: Singular blocks of the category O, and Soergel: n-Cohomology of simple highest weight modules on walls and purity. They give a formula for expressing singular parabolic KL-polynomials to the regular parabolic ones. – Rafael Mrđen Aug 27 '19 at 7:52
• I guess "his book" here means Humphreys's, but which one? I first looked at the reflection groups book, but it has no Section 9.7. – LSpice Aug 27 '19 at 10:07
• @LSpice: edited. – Rafael Mrđen Aug 27 '19 at 10:11
• Clickable: Irving - Singular blocks of the category $\mathscr O$ I II; Soergel - $\mathfrak n$-cohomology of simple highest weight modules on walls and purity. – LSpice Aug 27 '19 at 13:01