4
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What is the story for singular non-integral weights? Is it the same formula? $\endgroup$ – James Cheung Aug 27 at 7:38
  • $\begingroup$ 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. $\endgroup$ – Rafael Mrđen Aug 27 at 7:52
  • $\begingroup$ 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. $\endgroup$ – LSpice Aug 27 at 10:07
  • 1
    $\begingroup$ @LSpice: edited. $\endgroup$ – Rafael Mrđen Aug 27 at 10:11
  • $\begingroup$ Clickable: Irving - Singular blocks of the category $\mathscr O$ I II; Soergel - $\mathfrak n$-cohomology of simple highest weight modules on walls and purity. $\endgroup$ – LSpice Aug 27 at 13:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.