The Unreasonable Effectiveness of the Affine Hecke Algebra

理解对称性是一项非常有利可图的工作。例如，如果反射对称可用，则计算可以减半：只需要对象的一半就可以完全理解它。相反，对称性的缺乏也会限制一种材料或物体是否具有某种属性。这个项目的总体主题是通过研究某些对称组如何转换一维、二维、三维或更高维度的空间来更好地理解它们。当使用基于实数的代数空间时，这是众所周知的，但当在模数情况下工作时，即当实数被依赖于固定素数的有限数系统取代时，这一点要复杂得多。模算术在计算上很容易，而且非常有用；例如，它构成了编码理论和密码学中现代算法的基础。然而，理解和分类对称如何在模的情况下变换代数空间的问题比对于实数更困难，也没有被很好地理解。广义地说，这个项目将研究模情形下的代数空间的对称性。更准确地说，许多约化连通代数群的表示理论都是用仿射Hecke代数编码的。仿射Hecke代数有两种几何解释：仿射旗簇上的等变可构造层和Steinberg簇上的等变凝聚层。已知这两种解释在处理特征零滑轮时是等价的。在模块化的情况下，这一点尚不清楚。然而，如果它真的成立，人们可能会预计，相关方面的某些结构尚未定义。PI将定义凝聚层派生范畴上类似于Springer分解的奇异t-结构，并将为交上同调层而发展的强大理论推广到凝聚环境。PI还将为Satake类别定义一个Iwahori-Whittaker模型。最后，PI将显式地构造混合仿射Hecke范畴中的一些中心层(那些与倾斜表示相关联的层)。这个项目将加强目前对仿射黑克代数的绝对理解，这将有助于它在模表示理论中的应用。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Understanding symmetry is a very profitable endeavor. For instance, if reflective symmetry is available, then computations can be cut in half: only half of the object is needed to understand it fully. Conversely, a lack of symmetry can also limit whether a material or object can possess a certain property. The overall theme of this project is to better understand certain groups of symmetries by studying how they transform a one, two, three, or higher dimensional space. This is well-understood when working with algebraic spaces based on the real numbers, but much more complicated when working in the modular case, that is, when real numbers are replaced by a finite number system that depends on a fixed prime number. Modular arithmetic is computationally easy and very useful; for instance, it forms the basis of modern algorithms in coding theory and cryptography. However, the problem of understanding and classifying how symmetries transform algebraic spaces in the modular case is more difficult than for real numbers and is not as well-understood. Broadly speaking, this project will study symmetries of algebraic spaces in the modular case. More precisely, much of the representation theory of a reductive, connected algebraic groups is encoded in affine Hecke algebras. Affine Hecke algebras have two geometric interpretations: equivariant constructible sheaves on the affine flag variety and equivariant coherent sheaves on the Steinberg variety. These two interpretations are known to be equivalent when working with characteristic zero sheaves. In the modular case, this is not yet known. However, if it does hold, one would expect certain structures on the coherent side that have yet to be defined. The PI will define exotic t-structures on derived categories of coherent sheaves for varieties similar to the Springer resolution and extend the powerful theory developed for intersection cohomology sheaves to the coherent setting. The PI will also define an Iwahori--Whittaker model for the Satake category. Finally, the PI will explicitly construct some central sheaves (those associated to tilting representations) in the mixed affine Hecke category. This project will strengthen the current categorical understanding of the affine Hecke algebra which, will aid its application to modular representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.