Representations of Rational Cherednik Algebras in Positive Characteristic

1. Background
In 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. Aims
An 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.