## Representations of rational Cherednik algebras: Koszulness and localisation

##### Abstract

An algebra is a typical object of study in pure mathematics. Take a collection of numbers (for
example, all whole numbers or all decimal numbers). Inside, you can add and multiply, but
with respect to these operations different collections can behave differently. Here is an example
of what I mean by this. The collection of whole numbers is called Z. Starting anywhere in Z
you can get to anywhere else by adding other members of the collection: 9 + (-3) + (-6) = 0.
This is not true with multiplication; to get from 5 to 1 you would need to multiply by 1/5 and 1/5
doesn’t exist in the restricted universe of Z. Enter R, the collection of all numbers that can be
written as decimals. Now, if you start anywhere—apart from 0—you can get to anywhere else
by multiplying by members of R—if you start at zero you’re stuck there.
By adjusting what you mean by ‘add’ and ‘multiply’, you can add and multiply other
things too, like polynomials, transformations or even symmetries. Some of these collections
look different, but behave in similar ways and some look the same but are subtly different. By
defining an algebra to be any collection of things with a rule to add and multiply in a sensible way,
all of these examples (and many more you can’t imagine) can be treated in general. This is
the power of abstraction: proving that an arbitrary algebra, A, has some property implies that
every conceivable algebra (including Z and R) has that property too.
In order to start navigating this universe of algebras it is useful to group them together
by their behaviour or by how they are constructed. For example, R belongs to a class called
simple algebras. There are mental laboratories full of machinery used to construct new and
interesting algebras from old ones. One recipe, invented by Ivan Cherednik in 1993, produces
Cherednik algebras.
Attached to each algebra is a collection of modules (also called representations). As shadows
are to a sculpture, each module is a simplified version of the algebra, with a taste of its internal
structure. They are not algebras in their own right: they have no sense of multiplication, only
addition. Being individually simple, modules are often much easier to study than the algebra
itself. However, everything that is interesting about an algebra is captured by the collective
behaviour of its modules. The analogy fails here: for example, shadows encode no information
about colour. Sometimes the interplay between its modules leads to subtle and unexpected
insights about the algebra itself.
Nobody understands what the modules for Cherednik algebras look like. One first step is
to simplify the problem by only considering modules which behave ‘nicely’. This is what is
referred to as category O. Being Koszul is a rare property of an algebra that greatly helps to
describe its behaviour. Also, each Koszul algebra is mysteriously linked with another called its
Koszul dual. One of the main results of the thesis is that, in some cases, the modules in category
O behave as if they were the modules for some Koszul algebra. It is an interesting question to
ask, what the Koszul dual might be and what this has to do with Cherednik’s recipe.
Geometers study tangled, many-dimensional spaces with holes. In analogy with the algebraic
world, just as algebraists use modules to study algebras, geometers use sheaves to study
their spaces. Suppose one could construct sheaves on some space whose behaviour is precisely
the same as Cherednik algebra modules. Then, for example, theorems from geometry about
sheaves could be used to say something about Cherednik algebra modules. One way of setting
up this analogy is called localisation. This doesn’t always work in general. The last part of the
thesis provides a rule for checking when it does.