I am studying the BGG category $\mathcal{O}$. There is one problem I couldn't get out of my mind and I don't know enough to find a good reference.

Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $0$, and $\mathfrak{b}$ a Borel subalgebra of $\mathfrak{g}$ containing a Cartan subalgebra $\mathfrak{h}$. For $\lambda,\mu\in\mathfrak{h}^*$, with $\lambda$ being a regular integral weight, we can use Kazhdan-Lusztig polynomials to compute the composition factor multiplicity $\big[M(\lambda):L(\mu)\big]$, where $M(\lambda)$ is the Verma module with $\mathfrak{b}$-highest weight $\lambda$ and $L(\mu)$ is the simple module with $\mathfrak{b}$-highest weight $\mu$.

Do we have a similar formulation for non-integral weights? What about irregular weights? What if the weight $\lambda$ is both non-integral and irregular?

I greatly appreciate both references and comments on how to approach these situations.

  • 2
    $\begingroup$ For the theory (and references) up to about 2008, see the survey in Chapter 8 of my AMS text (and revisions on the AMS bookpage), especially 8.8. But note that Soergel's work has led recently to an alternative, more algebraic, derivation of the KL formula. Note too that the original formulation in the 1979 KL paper deals especially with the important special case of the 1-block in the category $\mathcal{O}$. $\endgroup$ – Jim Humphreys Feb 18 '17 at 18:25

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