5 research outputs found
Suzuki functor at the critical level
In this paper we define and study a critical-level 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 Calogero-Moser space into the space of opers on
the punctured disc. We give a partial geometric description of this embedding.Comment: Some changes in the presentation, a few minor mistakes correcte
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 and Hilbert schemes. The second problem concerns the critical-level 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 Khovanov-Lauda-Rouquier 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 Etingof-Ginzburg 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 q-hook 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 critical-level 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 Calogero-Moser 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 Brundan-Kleshchev-Rouquier isomorphism for the affine q-Schur algebra due to Miemietz and Stroppel
Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold Khovanov-Lauda-Rouquier algebras
Generalized Schur-Weyl dualities for quantum affine symmetric pairs and orientifold KLR algebras
We define a boundary analogue of the Kang-Kashiwara-Kim-Oh generalized
Schur-Weyl dualities between quantum affine algebras and
Khovanov-Lauda-Rouquier (KLR) algebras. Let be a complex simple
Lie algebra and the corresponding quantum affine algebra. We
construct a functor between finite-dimensional modules
over a quantum symmetric pair of affine type and an orientifold KLR (KLR) algebra arising from a
framed quiver with a contravariant involution, whose nodes are indexed by
finite-dimensional -modules. With respect to the
Kang-Kashiwara-Kim-Oh construction, our combinatorial model is further enriched
with the poles of the trigonometric K-matrices (that is trigonometric solutions
of a generalized reflection equation) intertwining the action of
on finite-dimensional -modules. By
construction, is naturally compatible with the
Kang-Kashiwara-Kim-Oh functor in that, while the latter is a functor of
monoidal categories, is a functor of module categories.
Relying on an isomorphism between suitable completions of KLR algebras and
affine Hecke algebras of type , we prove that
recovers the Schur-Weyl dualities due to Fan-Lai-Li-Luo-Wang-Watanabe in
quasi-split type . Finally, we construct spectral K-matrices for
orientifold KLR algebras, yielding a meromorphic braiding on its category of
finite-dimensional representations. We prove that, in the case of the quiver with no fixed points and no framing, the functor
is exact, factors through a suitable localization, and
takes values in a boundary analogue of the Hernandez-Leclerc category.Comment: 61 page
Representations of orientifold Khovanov-Lauda-Rouquier algebras and the Enomoto-Kashiwara algebra
We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier
algebras, depending on a quiver with an involution and a framing. Their
representation theory is related, via a Schur-Weyl duality type functor, to
Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to
highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our
first main result is a new shuffle realization of these highest weight modules
and a combinatorial construction of their PBW and canonical bases in terms of
Lyndon words. Our second main result is a classification of irreducible
representations of orientifold KLR algebras and a computation of their global
dimension in the case when the framing is trivial.Comment: 33 pages, small changes in the introduction, some references adde