Classically, the notion of total positivity referred to matrices all of whose minors had positive determinants. Lusztig generalized this notion substantially ([L1],[L2],[L3]) introducing the nonnegative part of an arbitrary reductive group, as well as the nonnegative part of a flag variety. Lusztig proved that the latter is always contractible and it has been conjectured to always be homeomorph...

Let G be a simple complex classical Lie group with Lie algebra g of rank n. We show that the coefficient of degree k in the Lusztig q-analogue K λ,μ(q) associated to the fixed partitions λ and μ stabilizes for n sufficiently large. As a consequence, we obtain the stabilization of the dimensions in the Brylinski-Kostant filtration associated to any dominant weight. We then introduce, for each pa...

where x ∈ Ŝk is minimal such that ν = λ.x −1 satisfies νi < νi+1 for i = 1, 2 . . . k−1 and νi−νk ≥ 1−k−n, and μ = λ.xy. This conjecture is proved by Kazhdan-Lusztig [KL] and Kashiwara-Tanisaki [KT]. The proof relies on an equivalence between the category of finite-dimensional Uǫ(slk)-modules and a category of negative-level representations of the affine algebra ŝlk which are integrable with re...

Let R be a root system. Let W be the corresponding affine Weyl group, and let Ŵ be an extended affine Weyl group. Let H (respectively Ĥ) be the corresponding Hecke algebras. George Lusztig defined an asymptotic version of the Hecke algebra, the ring J , see [10]. By definition the ring J is a direct sum J = ⊕ c Jc where summation is over the set of two-sided cells in the affine Weyl group. Furt...

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 th...

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description the -functions and Kazhdan-Lusztig cells these groups.