Multiplicity of the adjoint representation in simple quotients of the enveloping algebra of a simple Lie algebra

Abstract:

Let $\mathfrak {g}$ be a complex simple Lie algebra, $\mathfrak {h}$ a Cartan subalgebra and $U(\mathfrak {g})$ the enveloping algebra of $\mathfrak {g}$. We calculate for each maximal two-sided ideal ${J_{\max }}(\lambda ):\lambda \in {\mathfrak {h}^{\ast }}$ of $U(\mathfrak {g})$ the number of times the adjoint representation occurs in $U(\mathfrak {g})/{J_{\max }}(\lambda )$. This is achieved by reduction via the Kazhdan-Lusztig polynomials to the case when $\lambda$ lies on a corner, i.e. is a multiple of a fundamental weight. Remarkably in this case one can always present $U(\mathfrak {g})/{J_{\max }}(\lambda )$ as a (generalized) principal series module and here we also calculate its Goldie rank as a ring which is a question of independent interest. For some of the more intransigent cases it was necessary to use recent very precise results of Lusztig on left cells. The results are used to show how a recent theorem of Gupta established for "nonspecial" $\lambda$ can fail if $\lambda$ is singular. Finally we give a quite efficient procedure for testing if an induced ideal is maximal.