The composition series of modules induced from Whittaker modules
Let \({\mathfrak g}\) be a semisimple complex Lie algebra and \({\mathfrak n}\) the nilradical of a Borel subalgebra. Write \(U=U({\mathfrak g})\) for the enveloping algebra of \({\mathfrak g}\) and \(Z\) for its center. Choose \(\chi\) a maximal ideal in \(Z\) and \(f\) a character of \({\mathfrak n}\). Denoting by \({\mathbb C}_f\) the corresponding one dimensional representation of \({\mathfrak n}\), consider the \({\mathfrak g}\)-module \[ Y(\chi, f) = U/\chi U \otimes_{U({\mathfrak n})} {\mathbb C}_f. \] Such modules are of finite length. The problem studied in this paper is to compute the composition factors, with multiplicities, of \(Y(\chi, f)\). The problem is solved completely for integral \(\chi\) and partially for other \(\chi\) as well. The main idea is to reduce the problem to the Kazhdan-Lusztig conjectures describing the composition series of Verma modules. This is accomplished by making use of results of McDowell and proving an equivalence of certain natural categories.
scientific article; zbMATH DE number 1129395