| Résume | Affine Hecke algebras are central objects in representation theory. While their equal parameter theory is well understood using geometric methods, the unequal parameter case is considerably more subtle. In this talk, I will discuss a new geometric approach to affine Hecke algebras with unequal parameter which exploits certain small characteristic phenomena.
We will start by reviewing the seminal work of Kazhdan-Lusztig who gave a geometric construction of affine Hecke algebras with equal parameters via equivariant K-theory and the work of Kato who extended this to the unequal parameter setting in type C. Building on these ideas, I will then present a new geometric construction that realizes affine Hecke algebras with unequal parameters uniformly across all types. This framework also leads to a classification of their irreducible representations.
Finally, I will explain how these geometric ideas can be adapted to produce a diagrammatic categorification of Hecke algebras with unequal parameters in both finite and affine types, yielding new and interesting (p-)canonical bases. |
