Title:

Rational Cherednik algebras, quiver Schur algebras and cohomological Hall algebras

This thesis is devoted to three interrelated problems in representation theory. The first problem concerns the combinatorial aspects of the connection between rational Cherednik algebras at t=0 and Hilbert schemes. The second problem concerns the criticallevel limit of the Suzuki functor, which connects the representation theory of affine Lie algebras to that of rational Cherednik algebras. The third problem concerns the properties of certain generalizations of KhovanovLaudaRouquier algebras, called quiver Schur algebras, and their relationship to cohomological Hall algebras. Let us describe our results in more detail. In chapter 3, we study the combinatorial consequences of the relationship between rational Cherednik algebras of type G(l,1,n), cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct C*fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. We give a combinatorial description of the bijections between C*fixed points induced by the EtingofGinzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of a well known combinatorial identity, called the qhook formula. We also explain the connection between our results and Bezrukavnikov and Finkelberg's, as well as Losev's, proofs of Haiman's wreath Macdonald positivity conjecture. In chapter 4, we define and study a criticallevel generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings induced by the functor and deduce that every simple module over the rational Cherednik algebra lies in its image. Our homomorphism between the two centres gives rise to a closed embedding of the CalogeroMoser space into the space of opers on the punctured disc. We give a partial geometric description of this embedding. In chapter 5, we establish a connection between a generalization of KLR algebras, called quiver Schur algebras, and the cohomological Hall algebras of Kontsevich and Soibelman. More specifically, we realize quiver Schur algebras as algebras of multiplication and comultiplication operators on the CoHA, and reinterpret the shuffle description of the CoHA in terms of Demazure operators. We introduce ``mixed quiver Schur algebras" associated to quivers with a contravariant involution, and show that they are related, in an analogous way, to the cohomological Hall modules defined by Young. Furthermore, we obtain a geometric realization of the modified quiver Schur algebra, which appeared in a version of the BrundanKleshchevRouquier isomorphism for the affine qSchur algebra due to Miemietz and Stroppel.
