Kazhdan-Lusztig algorithms for nonlinear groups and applications to Kazhdan-Patterson lifting

We establish an algorithm to compute characters of irreducible Harish-Chandra modules for a large class of nonalgebraic Lie groups. (Roughly speaking the class of groups consists of those obtained as nonlinear double covers of linear groups in Harish-Chandra's class.) We then apply this theory to study a particular group (the universal cover of the real general linear group), and discover a symmetry of the character computations encoded in a character multiplicity duality. Using this duality theory, we reinterpret a kind of representation-theoretic Shimura correspondence for the general linear group geometrically, and find that it is dual to an analogous lifting for indefinite unitary groups. It seems likely that this example is illustrative of a general framework for studying similar correspondences.