On modules induced from Whittaker modules
type: Item
instance of: scholarly article
title: On modules induced from Whittaker modules
zbMATH Open document ID: 0584.17002
DOI: 10.1016/0021-8693(85)90044-4
author: Edward McDowell
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: Let \(\mathfrak g\) be a semisimple Lie algebra, and \(\mathfrak n\) a maximal nilpotent subalgebra. The irreducible modules for \(\mathfrak g\) containing a vector annihilated by \(\mathfrak n\) are called irreducible highest weight modules. They can be parametrized in a simple way, and exhibited as quotients of certain universal modules (Verma modules). Most Verma modules are irreducible, and all have finite composition series. The present paper considers (more generally) irreducible modules on which \(\mathfrak n\) acts in a locally finite way. Such a module \(M\) is generated by a vector \(v\) on which \(\mathfrak n\) acts by a character \(c\). If \(c\) is zero, \(M\) is a highest weight module. If \(c\) is generic, \(M\) is a ``Whittaker module'', as studied by \textit{B. Kostant} [Invent. Math. 48, 101--184 (1978; Zbl 0405.22013)]. The general case is a combination: the universal modules are induced from Whittaker modules on parabolic subalgebras. Analogues of the basic facts for Verma modules (mentioned above) are established. There is no attempt to extend the results of Bernstein, Gelfand, and Gelfand to this case.
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 3935263
zbMATH Keywords: maximal nilpotent subalgebra
zbMATH Keywords: Verma modules
zbMATH Keywords: irreducible modules
zbMATH Keywords: highest weight module
zbMATH Keywords: Whittaker module
zbMATH Keywords: parabolic subalgebras
reviewed by: David A. jun. Vogan
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90044-4
OpenAlex ID: W2083845801
cites work: Structure of representations generated by vectors of highest weight
cites work: Primitive ideals in enveloping algebras
cites work: Moduln mit einem höchsten Gewicht
cites work: On Whittaker vectors and representation theory
cites work: Structure of certain induced representations of complex semisimple Lie algebras
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description: scientific article; zbMATH DE number 3935263