Whittaker modules associated with highest weight modules
Let G be a connected complex semisimple Lie group with Lie algebra \({\mathfrak g}\). Let n be the nilradical of a Borel subalgebra of \({\mathfrak g}\) and m the nilradical of an arbitrary parabolic subalgebra. This paper is concerned with the structure and realization of various kinds of Whittaker modules for G. The basic idea is to generalize classical results of Kostant for n-Whittaker modules to m-Whittaker ones. (Some of the preliminary work in this direction had already been done by Lynch.) The author gives necessary and sufficient conditions for the finite- dimensionality of the space of Whittaker vectors under a standard genericity assumption. He constructs an exact functor from the category of highest weight modules to that of Whittaker modules and shows that it commutes with the translation functors. This functor leads to a natural embedding of certain important Whittaker modules into duals of Verma modules. Taking a slightly different tack, one might also try to find Whittaker submodules of various completions of (usually simple) highest weight modules; the author shows how to do this for Gevrey completions (defined by a certain natural family of seminorms). Finally, the author ties up his functor in a very nice way with the Casselman-Jacquet functors and Lie algebra cohomology. In subsequent work, the author studies the dimension of the space of Whittaker vectors when it is finite, and shows that for complex groups it transforms under the coherent continuation Weyl group action in the same way as a certain Goldie rank polynomial. \{A weaker version of Theorem 3.2.6 is given in a correction note [ibid. 61, No.3, 973 (1990)].\}
scientific article; zbMATH DE number 4179598
scientific article