On the Kazhdan-Lusztig conjectures

Perverse Sheaves and the Kazhdan-lusztig Conjectures

Brundan-kazhdan-lusztig and Super Duality Conjectures

We formulate a general super duality conjecture on connections between parabolic categories O of modules over Lie superalgebras and Lie algebras of type A, based on a Fock space formalism of their Kazhdan-Lusztig theories which was initiated by Brundan. We show that the Brundan-Kazhdan-Lusztig (BKL) polynomials for gl(m|n) in our parabolic setup can be identified with the usual parabolic Kazhda...

Positivity Conjectures for Kazhdan-lusztig Theory on Twisted Involutions: the Universal Case

Let (W,S) be a Coxeter system and let w → w∗ be an involution of W which preserves the set of simple generators S. Lusztig and Vogan have recently shown that the set of twisted involutions (i.e., elements w ∈ W with w−1 = w∗) naturally generates a module of the Hecke algebra of (W,S) with two distinguished bases. The transition matrix between these bases defines a family of polynomials Pσ y,w w...

On the quantum Kazhdan-Lusztig functor

One of the most exciting developments in representation theory in the recent years was the discovery of the Kazhdan-Lusztig functor [KL93a, KL93b, KL94a, KL94b], which is a tensor functor from the fusion category of representations of an affine Lie algebra to the category of representations of the corresponding quantum group, and is often an equivalence of categories. Informally speaking, this ...

Kazhdan-lusztig Cells

These are notes for a talk on Kazhdan-Lusztig Cells for Hecke Algebras. In this talk, we construct the Kazhdan-Lusztig basis for the Hecke algebra associated to an arbitrary Coxeter group, in full multiparameter generality. We then use this basis to construct a partition of the Coxeter group into the Kazhdan-Lusztig cells and describe the corresponding cell representations. Finally, we speciali...