From Kazhdan-Lusztig polynomials to exotic arc algebras
Kazhdan-Lusztig polynomials can be used to describe multiplicities in many interesting representation theoretic categories (e.g. BGG category O, certain categories of representations of Brauer algebras and Lie superalgebras). Computing these polynomials explicitly is difficult in general. In this talk we will focus on certain parabolic Kazhdan-Lusztig polynomials which can be determined easily using a diagram calculus. The diagrammatics can be used to define different versions of so-called arc algebras whose categories of modules are equivalent to the representation theoretic category of interest we started with.
The main part of this talk will be to explain the construction of a new such family of arc algebras related to the geometry of exotic Springer fibers. These algebraic varieties first appeared in Kato’s work on the Deligne-Langlands correspondence. If time permits, I will explain conjectural connections to categories of perverse sheaves, braid group actions on derived categories and link invariants.
Dr Arik Wilbert, University of Melbourne