On modules induced from Whittaker modules
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.
scientific article; zbMATH DE number 3935263
scientific article