This work is funded through the Professional Opportunities for Women in Research and Education (POWRE) program as a Research/Educational Enhancement Project. This project has two parts. The first is the development of a theory of Harish-Chandra modules for quantized enveloping algebras. These modules are important because of their impact on the study of Lie group representations. In the classical case, the Harish-Chandra modules of semisimple Lie algebras behave nicely when the action is restricted to maximal compact Lie subalgebras. To construct an analogous quantum theory, the investigator will analyze already known quantum analogs of these special Lie subalgebras and identify new ones. Next, basic results such as a quantum Harish-Chandra's subquotient theorem and a finiteness condition on isotypic components will be proved relative to these subalgebras. The second topic concerns generalizations of the classical theorem due to Chevalley: the invariants of the symmetric algebra of a semisimple Lie algebra under the adjoint group surjects onto the Weyl group invariants of the symmetrized Cartan subalgebra. The researcher will study a variation of this setup where the two groups are replaced with the smaller adjoint group and Weyl group associated to certain Lie subalgebras of a given semisimple Lie algebra. To what extent does the image of the first invariant subalgebra have finite codimension in the second? Answers will put recent solutions to a question of Wallach in a general context. The main technique is to exploit the so-called Letzter map, originally developed for quantized enveloping algebras.
