부산대학교 도서관

Let $L=N_+\oplus H\oplus N_-$ be a Kac\mhy Moody Lie algebra over a field $K$ of characteristic 0. For a Lie algebra $B$, let $U(B)$ denote the universal enveloping algebra, and let $U(B)^0$ be its augmentation ideal. For each $L$-module $M$ in the category $\scr O$ defined by \n V. Deodhar\en, \n O. Gabber\en and \n V. Kac\en [Adv. in Math. {\bf 45} (1982) no. 1, 92--116; MR0663417 (83i:17012)] a natural filtration by $L$-submodules $M_i$ of $M$ is associated with the set of elements of $M$ annihilated by $N_+$ in the following way. For $k=0,\ E_k(M)$ is defined as the set of $x$ in $M$ such that $e_ix=0$ for each generator $e_i$ of $N_+$, and for $k\ge1$, $E_k(M)$ is defined to be $U(N_-)E_k(M)$ for $k\ge0$. \n J. C. Jant\-zen\en [{\it Moduln mit einem höchsten Gewicht}, Lecture Notes in Math., 750, Springer, Berlin, 1979; MR0552943 (81m:17011)] conjectured that for finite-dimensional\break semisimple Lie algebras $M(\lambda)_i\subseteq M(\lambda)^i$ for all $i\ge0$, where $M(\lambda)$ is the Verma module of highest weight $\lambda$ in $H^*$ and where the $M(\lambda)^i$ comprise the Jantzen filtration of $M(\lambda)$. The author studies Jant\-zen's conjecture for $L$ of Kac\mhy Moody type and for a certain class of Verma modules that can be characterized by the coincidence of the two above-described filtrations. For $\mu$ in $H^*$, let $I(\mu)$ be the irreducible quotient of $M(\mu)$ and let $[M\colon I(\mu)]$ be the multiplicity of $I(\mu)$ in $M$. Assume that $L$ is symmetrizable. Then $M(\lambda)_i=M(\lambda)^i$ for all $i$ if and only if $[M(\lambda)\colon I(\mu)]=\dim\,\roman{Hom}_G (M(\mu),M(\lambda))$ for all $\mu$ in $H^*$. The proof, which uses the nondegenerate form on each $M(\lambda)^i/M(\lambda)^{i+1}$ associated with the construction of the Jantzen filtration and the generalized Casimir operator, yields a corollary that could be used to construct an inductive proof of the Jantzen conjecture, if a certain condition too technical to specify here could be established for some $i\ge0$. It would imply that $M(\lambda)_i\subseteq M(\lambda)^i$ entails this for $i+1$ also. For symmetrizable $L$, additional corollaries follow on the heredity of the Jantzen filtration and the relative filtration for embeddings.