Generalized Robinson-Schensted Algorithms

In Part I of this thesis, we locate a (conjecturally complete) set of unitary representations in the admissible dual of U(p, q). In a little more detail, Barbasch and Vogan have used the theory of Kazhdan-Lusztig cells to parametrize the irreducible Harish-Chandra modules with integral infinitesimal character in terms of their annihilators and associated varieties. Vogan has conjectured that the weakly fair cohomologically induced modules A,(A) exhaust the unitary dual of U(p, q) for the kinds of infinitesimal character that they can have. Here we compute the annihilators and associated varieties of these modules, thus locating them in the admissible dual. In particular, this determines all coincidences among these modules and gives their Langlands parameters. We conclude Part I with some evidence for the conjecture. In Part II, we interpret some of the combinatorics which arise in the Barbasch-Vogan parametrization in terms of the geometry of the generalized Steinberg variety. This leads to a study of geometric cells, which are exactly analogous to Kazhdan-Lusztig cells, except that one begins with a topological action of the complex Weyl group instead of the coherent continuation action. We compute the structure of geometric cells for type A real groups; more precisely, we compute Springer's generalized Robinson-Schensted algorithm for these groups, and compare the computation to the Barbasch-Vogan parametrization. Thesis Committee: Bertram Kostant George Lusztig David A. Vogan, Jr. (Thesis Advisor)