The Jantzen filtration of a certain class of Verma modules

Abstract:

Let $G = {N_ + } \oplus H \oplus {N_ - }$ be a Kac-Moody Lie algebra. For each $M$ in the category $\mathcal {O}$ of $G$-modules, there is a filtration ${({M_i})_{i \geq 0}}$ by $G$-submodules of $M$ naturally associated with the set $\left \{ {\upsilon \in M\left | {{N_ + }\upsilon } \right . = 0} \right \}$. If $G$ is symmetrizable and $M$ is a Verma module, ${M_i} = {M^i}$ for all $i$ if and only if $\left [ {M:L(\mu )} \right ] = \dim \operatorname {Hom}_G(M(\mu ),M)$ for all $\mu \in {H^ * }$ where ${({M^i})_{i \geq 0}}$ is the Jantzen filtration of $M$. The main tools used are the nondegenerate form on each ${M^i}/{M^{i + 1}}$ together with the $\Gamma$-operator of $G$.