Twisted cells for real reductive lie groups

Given a real reductive linear Lie group G, consider the nite set D of Langlands parameters corresponding to (innitesimal equivalence classes of) irreducible admissible representations of G having innitesimal character . We may construct a free Z[q; q􀀀1]-moduleMwith two bases ft g; fc g (respectively the \standard" and \irreducible" basis of M), each indexed by elements 2 D, and with change of basis given by the so-called Kazhdan-Lustig-Vogan polynomials. This module M admits an action of the Hecke algebra H(W), where W is the complex Weyl group of G. The action of H(W) on the irreducible basis of M allows us to dene a partial ordering on D; the equivalence classes C of D induced by this partial ordering are called cells of D. Now, let  be an involutive automorphism of G which preserves a xed Borel subgroup B  G. The action of  on G induces an action of  on both the set D and the complex Weyl group W, and we let D and W be the respective sets of xed points under this action. As above, we may construct a free Z[q; q􀀀1]-module M with two bases ft g; fc g (the \standard" and \irreducible" bases of M), each indexed by elements 2 D. The module M admits an action of the unequal parameter Hecke algebra H(W), and in the same fashion as above, the action of H(W) on the irreducible basis of M induces a partition of the set D into equivalence classes C called twisted cells. In this dissertation, we investigate the relationship between twisted and untwisted cells. In particular, we conjecture that each twisted cell C of D, when viewed as a subset of D, is completely contained in some untwisted cell C of D. We give partial results to support this conjecture in the case of an arbitrary real reductive linear Lie group G, and describe a method to prove the conjecture in the case of G = U(n; n).