Hecke algebras and representation theory

Hecke代数是Coxeter群的群代数的一种变形，在表示理论中占有重要地位。这些代数也许是通过对哈希-钱德拉尖头形式哲学之后的诱导表征的研究而首次崭露头角，但现在涉及该领域的许多方面及其与数学其他部分的相互作用。在p进群的研究中自然产生的一类重要的Hecke代数是附在仿射Coxeter群上的仿射Hecke代数。虽然与任意Coxeter群相关的Hecke代数具有类似于底层群的Coxeter结构的标准表示，但仿射Hecke代数的一个特殊特征是，它们也有一个完全不同的，尽管不太明显的表示，这是Bernstein首先发现的。这第二种表述与以下事实密切相关：仿射Coxeter群可以通过晶格实现为有限Coxeter群的扩展，而仿射Hecke代数既包含与相应有限Coxeter群相关的（有限）Hecke代数，也包含相应晶格的群代数。上面描述的两种表述是更丰富的几何结构的影子：Kazhdan和Lusztig的著名工作，对仿射Hecke代数的不可约表示进行了分类，表明Bernstein的表述反映了这样一个事实，即仿射Hecke代数可以被实现为斯坦伯格首先研究的一个变种的等变k理论。另一方面，Coxeter表示则反映了仿射Hecke代数可以被实现为仿射旗变体上一类可构造束的合适k群。Bezrukavnikov最近的工作表明，人们可以对这两个表示实现相同代数的事实进行分类：仿射标志变体上的可构造范畴与斯坦伯格变体上的等变相干束范畴之间存在自然等价的范畴。除了它内在的美之外，这个深度等价已经被用来证明非常重要的结果（例如，Achar-Rider对Mirkovic-Vilonen猜想的证明，证明了仿射旗帜上标准束的茎部不存在扭转）。此外，越来越明显的是，现在不仅要研究（仿射）赫克代数，而且要研究它们的分类类似物，即所谓的赫克范畴，这一点非常重要。在这样做的过程中，似乎很自然地要调查一个人能在多大程度上产生这些类别的“呈现”。这也与研究以Hecke代数为商的辫群的范畴作用密切相关。在这个方向上的任何结果的一个可能的应用将是在统一根上的量子群的表示。当取量子群的“晶体”积分形式时，对单位根的专门化产生具有大中心的代数，Backelin-Kremnizter和Tanisaki的工作表明，这种代数的表示理论与施普林格分辨率（特征0）的几何密切相关。这种关系在施普林格纤维的相干束类别上产生了有趣的t型结构，这应该与Bezrukavnikov和Mirkovic在正特征中产生的类似t型结构有关（在他们与Rumynin的开创性论文的基础上发展而来的工作）。在正特征设置中，已知这些t型结构是由编织群作用控制的，并且人们期望在量子情况下得到类似的结果。建立这样的结果应该允许人们在统一根量子群的表示理论和模理论之间建立一个新的直接桥梁。该项目属于EPSRC代数研究领域，与EPSRC几何与拓扑研究领域有联系。

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.