Hecke algebras and representation theory

Hecke代数是Coxeter群的群代数的变形，在表示论中起着重要的作用。这些代数也许首先上升到突出通过研究诱导陈述以下哈里什-钱德拉的哲学尖点形式，但现在触及许多方面的领域及其相互作用与其他部分的mathematics.An重要类的赫克代数自然产生的研究p进群是仿射赫克代数重视仿射考克斯特群。虽然与任意Coxeter群相关联的Hecke代数具有类似于基础群的Coxeter结构的标准表示，但仿射Hecke代数的一个特殊特征是它们也具有完全不同的表示，尽管不太明显，但首先由伯恩斯坦发现。这第二个介绍是密切相关的事实，即仿射考克斯特群可以实现为一个有限的考克斯特群的一个扩展的格和仿射赫克代数包含（有限）赫克代数相关联的相应有限考克斯特群和相应的格的群代数。上述两个介绍原来是更丰富的几何结构的阴影：著名的工作Kazhdan和Lusztig，其中分类的不可约表示的仿射Hecke代数，表明伯恩斯坦介绍是一个反映的事实，即仿射Hecke代数可以实现为等变K理论的一个品种首先研究斯坦伯格。另一方面，Coxeter表示反映了仿射Hecke代数可以被实现为仿射旗簇上的一类可构造层的适当K-群的事实。Bezrukavnikov最近的工作表明，人们可以归类的事实，这两个介绍实现了相同的代数：有一个自然的范畴之间的范畴之间的可构造范畴的仿射旗品种和范畴的等变相干层的斯坦伯格品种。除了其内在的美，这种深度等价已经被用来证明高度非平凡的结果（例如Achar-Rider证明了Mirkovic-Vilonen猜想关于仿射旗簇上标准层的茎中不存在扭转）。此外，越来越清楚的是，现在不仅要研究（仿射）Hecke代数，而且要研究它们的范畴类似物，即所谓的Hecke范畴。在这样做的时候，似乎很自然地要调查在多大程度上可以产生这些类别的“介绍”。这也与研究辫子群的范畴作用密切相关，Hecke代数是辫子群的一个分支，这方面的任何结果都可能应用于量子群在单位根上的表示。当取量子群的“结晶”积分形式时，特化到单位根会产生一个具有大中心的代数，而Backelin-Kremnizter和Tanisaki的工作表明，这种代数的表示论与Springer归结的几何密切相关（特征为0）。这种关系在斯普林格纤维上的相干层范畴上产生了有趣的t结构，这应该与Bezrukavnikov和Mirkovic（在他们与Rumynin的开创性论文中发展起来的工作）以正特征产生的类似t结构有关。在正特征设定中，这些t-结构被认为是由辫群作用控制的，并且在量子情况下也可以预期类似的结果。建立这样的结果应该允许一个新的直接桥梁之间的表示理论的量子群在根的单位和modular theory.This project falls福尔斯在EPSRC代数研究领域与连接到EPSRC几何与拓扑研究领域。

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.