Tensor Products on Category O and Kostant's Problem

Abstract: This thesis consists of a summary and three papers, concerning some aspects of representation theory for complex finite dimensional semi-simple Lie algebras with focus on the BGG-category O.Paper I is motivated by the many useful properties of functors on category O given by tensoring with finite dimensional modules, such as projective functors and translation functors. We study properties of functors on O given by tensoring with arbitrary (possibly infinite dimensional) modules. Such functors give rise to a faithful action of O on itself via exact functors which preserve tilting modules, via right exact functors which preserve projective modules, and via left exact functors which preserve injective modules.Papers II and III both deal with Kostant's problem. In Paper II we establish an effective criterion equivalent to the answer to Kostant's problem for simple highest weight modules, in the case where the Lie algebra is of type A. Using this, we derive some old and new results which answer Kostant's problem in special cases. An easy sufficient condition derived from this criterion using Kazhdan-Lusztig combinatorics allows for a straightforward computational check using a computer, by which we get a complete answer for simple highest weight modules in the principal block of O for algebras of rank less than 5.In Paper III we relate the answer to Kostant's problem for certain modules to the answer to Kostant's problem for a module over a subalgebra. We also give a new description of a certain quotient of the dominant Verma module, which allows us to give a bound on the multiplicities of simple composition factors of primitive quotients of the universal enveloping algebra.