Hecke algebras and representation theory

Hecke algebras, which are a deformation of the group algebras of Coxeter groups, play an important role in representation theory. These algebras perhaps first rose to prominence through the study of induced representations following Harish-Chandra's philosophy of cusp forms, but now touch on many aspects of the field and its interactions with other parts of mathematics.
An important class of Hecke algebras naturally arising in the study of p-adic groups are the affine Hecke algebras attached to affine Coxeter groups. While Hecke algebras associated to arbitrary Coxeter groups have a standard presentation resembling the Coxeter structure of the underlying group, a special feature of affine Hecke algebras is that they also have a quite different, though less evident, presentation first discovered by Bernstein. This second presentation is closely related to the fact that affine Coxeter groups can be realised as an extension of a finite Coxeter group by a lattice and affine Hecke algebras contain both the (finite) Hecke algebra associated to the corresponding finite Coxeter group and the group algebra of the corresponding lattice.
The two presentations described above turn out to be shadows of richer geometric structures: The celebrated work of Kazhdan and Lusztig, which classified the irreducible representations of affine Hecke algebras, showed that the Bernstein presentation is a reflection of the fact that the affine Hecke algebra can be realized as the equivariant K-theory of a variety first studied by Steinberg. The Coxeter presentation, on the other hand, reflects the fact that the affine Hecke algebra can be realized as a suitable K-group of a category of constructible sheaves on the affine flag variety. More recent work of Bezrukavnikov has shown that one can categorify the fact that these two presentations realize the same algebra: there is a natural equivalence of categories between the constructible category on the affine flag variety and the category of equivariant coherent sheaves on the Steinberg variety.
Aside from its intrinsic beauty, this deep equivalence has been used to prove highly non-trivial results (for example Achar-Rider's proof of the Mirkovic-Vilonen conjecture on the absence of torsion in the stalks of standard sheaves on the affine flag variety). It has moreover become increasingly clear that it is important now to study not just (affine) Hecke algebras, but also their categorical analogues, the so-called Hecke categories. In doing so, it seems natural to investigate to what extent one can produce "presentations" of such categories. This is also closely related to studying categorical actions of the braid groups of which Hecke algebras are a quotient.
A possible application of any results in this direction would be to representations of quantum groups at a root of unity. When taking the "crystalline" integral form of a quantum group, the specialization to a root of unity yields an algebra with large centre, and work of Backelin-Kremnizter and Tanisaki has shown that the representation theory of such algebras is intimately related to the geometry of the Springer resolution (in characteristic 0). This relationship produces interesting t-structures on the categories of coherent sheaves on Springer fibres, which should be related to similar t-structures produced in positive characteristic by Bezrukavnikov and Mirkovic (in work that developed from their seminal paper with Rumynin). In the positive characteristic setting these t-structures are known to be controlled by a braid group action, and one expects similar results in the quantum case. Establishing such a result should allow one to construct a new direct bridge between the representation theory of quantum groups at roots of unity and the modular theory.
This project falls within the EPSRC Algebra research area with connections to the EPSRC Geometry & Topology research area.