Monodromy of Partial Kz Functors for Rational Cherednik Algebras

1.1. Shan has proved that the categories Oc(Wn) for rational Cherednik algebras of type Wn = W (G(`, 1, n)) = Snn(μ`) with n varying, together with decompositions of the parabolic induction and restriction functors of Bezrukavnikov-Etingof, provide a categorification of an integrable s̃le Fock space representation F(m), [18]. The parameters m ∈ Z` and e ∈ N ∪ {∞} arise from the choice of parameters c for the rational Cherednik algebra. This categorification gives rise to a crystal structure on the set of irreducible rational Cherednik algebra representations that belong to category Oc; it is isomorphic to the crystal introduced by Jimbo-Misra-Miwa-Okado, [11]. The works of many authors, including Kleshchev, Brundan, Lascoux-Leclerc-Thibon, Ariki, Grojnowski-Vazirani, Grojnowski and Chuang-Rouquier, show that the categoriesHq(Wn) –mod for Hecke algebras of type Wn with n varying, together with decompositions of the parabolic induction and restriction functors, provide a categorification of an irreducible integrable s̃le-representation L(Λ), [6]. The weight Λ arises from the choice of parameters q for the Hecke algebra. This gives rise to a crystal structure on the set of irreducible Hecke algebra representations. The Fock space is substantially more interesting than the representation L(Λ). It is not irreducible, and in fact has an infinite number of non-zero isotypic components. This reducibility reveals itself through distinct canonical bases one can define on F(m), each of which produces a corresponding crystal. Nonetheless for each n ∈ N there is an exact functor KZn : Oc(Wn)→ Hq(Wn) –mod, [10], which intertwines the parabolic induction and restriction functors for Cherednik algebras and Hecke algebras and produces a compatibility between the corresponding crystals: the component of the Cherednik crystal containing the irreducible representation of W0 = {1} is isomorphic to the Hecke crystal.