Parity Sheaves and the Hecke Category

Abstract

IntroductionOne of the first theorems of representation theory is Maschke's theorem: any representation of a finite group over a field of characteristic zero is semi-simple. This theorem is ubiquitous throughout mathematics. (We often use it without realising it; for example, when we write a function of one variable as the sum of an odd and an even function.) The next step is Weyl's theorem: any finite-dimensional representation of a compact Lie group is semi-simple 1 . It is likewise fundamental: for the circle group Weyl's theorem is closely tied to the theory of Fourier series.Beyond the theorems of Maschke and Weyl lies the realm of where semi-simplicity fails. Non semi-simple phenomena in representation theory were first encountered when studying the modular (i.e. characteristic p) representations of finite groups. This theory is the next step beyond the classical theory of the character table, and is important in understanding the deeper structure of finite groups. A second example (of fundamental importance throughout mathematics from number theory to mathematical physics) occurs when studying the infinite-dimensional representation theory of semi-simple Lie groups and their p-adic counterparts.Throughout the history of representation theory, geometric methods have played an important role. Over the last forty years, the theory of intersection cohomology and perverse sheaves has provided powerful new tools. To any complex reductive group is naturally associated several varieties (e.g. unipotent and nilpotent orbits and their closures, the flag variety and its Schubert varieties, the affine Grassmannian and its Schubert varieties . . . ). In contrast to the group itself, these varieties are often singular. The theory of perverse sheaves provides a collection of constructible complexes of sheaves (intersection cohomology sheaves) on such varieties, and the "IC data" associated to intersection cohomology sheaves (graded dimensions of stalks, total cohomology, . . . ) appears throughout Lie theory.The first example of the power of this theory is the Kazhdan-Lusztig conjecture (a theorem of Beilinson-Bernstein and Brylinski-Kashiwara), which expresses the character of a simple highest weight module over a complex semi-simple Lie algebra in terms of IC data of Schubert varieties in the flag variety. This theorem is an important first step towards understanding the irreducible representations of semisimple Lie groups. A second example is Lusztig's theory of character sheaves, 1 Weyl first proved his theorem via integration over the group to produce an invariant Hermitian form. To do this he needed the theory of manifolds. One can view his proof as an early appearance of geometric methods in representation theory.

Localized calculus for the Hecke category

“…[RW15, AMRW17, EL, HW15, Wil17, BCH20, EH17, GH17]) and higher representation theory. The reader is referred to the introduction to [EW16] as well as [Wil18] for a discussion of the origins of the Hecke algebra and category, as well as its significance in geometric and higher representation theory.…”

A homomorphism between Bott-Samelson bimodules

“…Here, the Hecke category means a categorification of the Hecke algebra of Coxeter groups. One can find the importance of the Hecke category in representation theory in Williamson's survey [Wil18].…”

Reducible characteristic cycles of Harish-Chandra modules for U(p,q) and the Kashiwara–Saito singularity

“…This remained open until Kashiwara-Saito [KS] found a surprising example in G = GL(8, C). The relevant singularity controlling this example was an an eight-dimensional subvariety of C 16 (recalled in (3.24) below) and was called the Kashiwara-Saito singularity in [Wi2,Example 1.14]. The reducibility example of Kashiwara-Saito had the property that the two irreducible components had different moment map images in the dual g * of the Lie algebra of G. For this reason, results of Joseph [J2] show that the reducibilty could not be detected by considering associated varieties in g * of simple highest weight modules for g (in the sense of [J1]).…”