Affine Hecke algebras, canonical bases and a derived Deligne-Langlands correspondence

Thesis

This thesis makes several contributions to the representation theory of affine Hecke algebras and related topics:

• We show that there is an equivalence of triangulated categories relating certain (dg-)modules over affine Hecke algebras with certain categories of constructible sheaves. This can be viewed as a derived version of the celebrated Deligne- Langlands correspondence.

• We provide a new algorithm that computes the canonical basis in any quantum group of finite ADE type or equivalently the stalks of perverse sheaves on a corresponding moduli space of quiver representations. In type A, this also encodes the composition multiplicities of the standard modules for the affine Hecke algebra of GLn and we show further how this algorithm can be extended to compute the dimensions of the simple modules.

• We construct a geometric realization of any affine Hecke algebra of simple type with two unequal parameters via equivariant algebraic K-theory. This can be viewed as an extension of a classical theorem due to Kazhdan-Lusztig and Ginzburg which provides such a realization in the equal parameter setting.

Authors: Antor, J

• DOI: 10.5287/ora-gproer7wj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Geometric Representation Theory