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.
