Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Anthony Joseph
Journal: Trans. Amer. Math. Soc. 316 (1989), 447-491
MSC: Primary 17B35; Secondary 22E47
DOI: https://doi.org/10.1090/S0002-9947-1989-1020500-3
MathSciNet review: 1020500
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B35, 22E47

Retrieve articles in all journals with MSC: 17B35, 22E47


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-1020500-3
Article copyright: © Copyright 1989 American Mathematical Society