Representations of Rational Cherednik Algebras in Positive Characteristic

1. 1994年，Ivan Cherednik引入了双仿射Hecke代数（DAHA），并用它们证明了Macdonald常数项猜想。Cherednik代数及其分支在越来越多的领域变得非常有用。Cherednik代数的表示理论与代数几何、组合学、有限维代数、同调代数、可积系统、李论、非交换代数和q{微积分有很强的联系；它们被用来证实所有这些学科的猜想和回答问题。2000年，Pavel Etingof和Victor Ginzburg引入了有理Cherednik代数（RCA）作为Cherednik代数的有理退化，其表示理论得到了广泛的研究。在特征0中，这是一个非常活跃的研究领域。对特征p的了解要少得多，这使它成为一个非常丰富的研究领域。即使是最基本的问题，比如描述简单模块的特征，也只在少数情况下得到了回答。2. 表示理论的一个重要目标是描述简单的模块。它们就像所有模块的构建块，从某种意义上说，任何模块都可以由简单模块作为直接和或通过扩展构造而成。我的目的始终是描述有理Cherednik代数的简单模块。首先，我们构建一个称为Verma模块的大模块。Verma模块非常容易定义和描述。例如，所有Verma模块的字符都是显式已知的。每一个Verma模都有一个极大真子模，对这个子模进行商得到一个简单模。此外，每个简单模块都以这种方式出现。虽然我们知道它的存在，但通常没有对极大固有子模的显式描述。因此，没有对我们作为商得到的简单模的显式描述。“什么时候一个简单模块是有限维的，在这种情况下，它的维数是多少？”这个问题通常仍然没有定论。我的目的是尽可能精确地描述有理Cherednik代数的简单模，并且在尽可能多的情况下，当基础域具有正特征p时，对于有限反射群G和两个数值参数t和c，我们可以形成一个记为Ht；c(G)的有理Cherednik代数。它的Verma模，因此表示为L(T)的简单模，被G的不可约表示T额外参数化。我将重点讨论G的特殊情况；t;c;T和p在不同的子项目中。

1. BackgroundIn 1994, Ivan Cherednik introduced the double affine Hecke algebras (DAHA) and used them to prove the Macdonald constant term conjecture. Cherednik algebras and their offshoots have since become very useful in a growing number of areas. The representation theory of Cherednik algebras has strong connections to algebraic geometry, combinatorics, finite dimensional algebra, homological algebra, integrable systems, Lie theory, noncommutative algebra and q{calculus; they have been used to confirm conjectures and answer questions in all of these subjects.In 2000, Pavel Etingof and Victor Ginzburg introduced rational Cherednik algebras (RCA) as rational degenerations of Cherednik algebras, and their representation theory is the subject of extensive study. In characteristic 0, it is an extremely active area of research. Far less is known in characteristic p, which makes it a very bountiful area for investigation. Even basic questions, such as describing characters of simple modules, have only been answered in a few cases. 2. AimsAn important goal in representation theory is to describe simple modules. They act like the building blocks of all modules, in the sense that any module can be constructed from simple modules as a direct sum, or by extensions. My aim throughout is to describe the simple modules of rational Cherednik algebras. First we construct a large module called a Verma module. Verma modules are very easy to define and describe. For example, the characters of all Verma modules are known explicitly. Every Verma module has a maximal proper submodule, and if we quotient by this submodule we obtain a simple module. Moreover, every simple module arises in this way.Although we know it exists, there is generally no explicit description of the maximal proper submodule. Consequently, there is no explicit description of the simple module that we obtain as a quotient. The question "When is a simple module finite-dimensional and, in that case, what is its dimension?" generally remains open.Aim. My aim is to describe the simple modules of rational Cherednik algebras as precisely as possible, and in as many settings as possible, when the underlying field has positive characteristic p.For a finite reflection group G, and two numerical parameters t and c, we can form a rational Cherednik algebra denoted Ht;c(G). Its Verma modules, and consequently the simple modules denoted L(T ), are additionally parametrised by irreducible representations T of G. I will focus on particular special cases of G; t; c; T, and p in different subprojects.