Rational Cherednik algebras and link invariants

Motivated by homological mirror symmetry, Smith and Thomas tried to construct a link invariant considering the derived category of coherent sheaves on the Hilbert scheme of n points on the minimal resolution of the Klenian singularity of type A, and an object ℒ(n) thereof. The braid group acts on this category by spherical twists, so one obtains a braid invariant by taking the Ext between ℒ(n) and its image under the braid group action. Smith and Thomas proved that taking the plat closure of the braid, this cohomology does not produce a link invariant but is close to doing so, and they conjectured that, in order to fix the one knot relation that is not satisfied, one has to consider a deformation of the Hilbert scheme. In this thesis, we give a non-commutative approach to this problem: the commutative picture can be quantised by considering modules for the rational Cherednik algebra of cyclotomic type. This algebra gives a quantisation of the Hilbert scheme and there is a localisation theorem which allows one to work in the algebraic setting. In this context, the role of ℒ(n) turns out to be played by a certain module for the rational Cherednik algebra which we define for k=0. We then show that this module deforms to non-zero values of k. There is an action of the braid group on the derived category of category by twisting functors, which is defined at all deformation parameters, whereas the existence of the action on deformed Hilbert schemes in the commutative setting has not been rigorously established. We prove the analogue of the Smith-Thomas theorem, and conjecture that the braid invariant given by the algebraic analogue of the Smith-Thomas construction yields a link invariant for certain non-zero values of the deformation parameter.