The Unreasonable Effectiveness of the Affine Hecke Algebra

理解对称性是一项非常有益的奋进。例如，如果反射对称是可用的，那么计算可以减半：只需要一半的对象就可以完全理解它。相反，缺乏对称性也会限制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它们如何转换一维，二维，三维或更高维空间来更好地理解某些对称群。当处理基于真实的数的代数空间时，这是很容易理解的，但当处理模的情况时，即当真实的数被依赖于固定素数的有限数系统代替时，这要复杂得多。模运算在计算上很容易，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，在模的情况下，如何理解和分类对称如何变换代数空间的问题比真实的数更困难，也没有得到很好的理解。广义地说，这个项目将研究代数空间在模情况下的对称性。更确切地说，一个约化的连通代数群的表示理论的大部分都是在仿射赫克代数中编码的。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。这两种解释是已知的等效时，与特征零滑轮。在模块化的情况下，这还不知道。然而，如果它真的成立，人们会期望在连贯的一面有某些结构，这些结构还有待定义。PI将定义外来的t-结构的衍生类别的相干层的品种类似的施普林格决议，并扩展强大的理论开发的交叉上同调层的相干设置。PI还将为Satake类别定义Iwahori-Whittaker模型。最后，PI将在混合仿射Hecke范畴中显式地构造一些中心层（与倾斜表示相关的中心层）。该项目将加强目前仿射Hecke代数的分类理解，这将有助于其应用于模块化表示理论。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.