A module induced from a Whittaker module

Abstract:

In an earlier paper [On modules induced from Whittaker modules, J. Algebra 96 (1985)] we constructed a class of induced modules, over a finite-dimensional semisimple Lie algebra, which includes the Verma modules of Verma [Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968)] and the irreducible Whittaker modules of Kostant [On Whittaker vectors and representation theory, Invent. Math. 48 (1978)]. We proved that every module in this class has finite length and is irreducible most of the time. In this article we present a concrete example of this construction, over $\operatorname {sl} (3,C)$, showing that proper submodules can exist when the induced module is not a Verma module.