Let $\mathfrak{g}$ be a complex symmetrizable Kac-Moody algebra, with triangular decomposition $\mathfrak{n}^- \oplus \mathfrak{h} \oplus \mathfrak{n}^+$. Let $\lambda \in \mathfrak{h}^*$, and $M(\lambda)$ be the corresponding Verma module. Then $M(\lambda)$ admits a Jantzen filtration, as explained e.g. in Section 3.1 of Kubel's thesis.

When $\mathfrak{g}$ is of finite type, $\lambda = w \cdot \nu := w(\nu + \rho) - \rho$, where $\nu$ is a dominant integral weight and $\rho$ is half the sum of the positive roots, it is known that the subquotients of the Jantzen filtration are semisimple, and their isotypic multiplicities are given by coefficients of Kazhdan-Lusztig polynomials (see e.g. Chapter 8 of the book of Humphreys on Category $\mathcal{O}$).

My question is: what is known/expected about the semisimplicity and isotypic multiplicities of subquotients of the Jantzen filtration in infinite type? I would also be interested in what is known/expected in finite type for more general $\lambda$.

Thank you for your help, and please let me know if anything is unclear!



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