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... is a 1-faithful highest weight cover of B with associated order ≤2 and if ≤2 is a refinement of ≤ ′ , then C1 and C2 are equivalent highest weight covers. 5. Cherednik’s rational algebra We refer to =-=[Rou2]-=- for a survey of the representation theory of rational Cherednik algebras. 5.1. Category O. 5.1.1. Given H ∈ A, let αH ∈ V ∗ with H = ker αH and let vH ∈ V such that CvH is a WH-stable complement to H...

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...ble systems, combinatorics, and symplectic quotient singularities. They are also closely related to the Hilbert scheme Hilb n C 2 of points in the plane, a connection that was formalized in [GS1] and =-=[KR]-=-, where it was shown that one can describe Uc as a kind of noncommutative deformation of Hilb n C 2 . This was then used in [GS1, GS2] to apply the geometry of Hilb n C 2 and Haiman’s work on the n! c...

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...ric, and elliptic. The rational DaHa, which goes back as a special case to Drinfeld [3], has been actively studied in recent years by many authors (see Etingof-Ginzburg [5] and the review of Rouquier =-=[11]-=- for extensive references). The question addressed here is whether or not a reasonable notion of DaHa’s associated to the spin symmetric group of Schur [13] exists. For Coxeter groups, there is a stan...

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...g the blocks of the Ariki-Koike algebra in a special case. 1. Introduction Let W be the complex reflection group G(m, 1, n) = Zm≀Sn with its natural n–dimensional representation h. It is suggested in =-=[Rou05]-=- that it may be possible to give a complete description of the structure of category O for the rational Cherednik algebra of the group W, using generalisations of the methods of [Gor03], [BEG03b], [GG...

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...sponds to a W–equivariant vector bundle on h reg with a flat connection ∇. The horizontal sections of ∇ define a system of differential equations on h reg which, by a process described in [BMR98] and =-=[Rou]-=-, give a monodromy representation of π1(h reg /W). By definition, KZ(M) is the monodromy representation of π1(h reg /W) associated to M. 74.7. By [BMR98, 4.12] and [GGOR03, Section 5.25], the monodro...

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...5). The key to the argument will be the twisting of Dh(λ ) modules by these line bundles. We formulate this twisting by using the Hamiltonian reduction definition of differential operators; following =-=[KR]-=-. In particular, as above (in section 5), we have equivalences of categories Modqc(Dh(λ ))→̃ModB,qcλ (Dh(T ∗G)) and Modcoh(Dh(λ ))→̃ModB,cohλ (Dh(T ∗G)) On T ∗G, we have, if V is any finite dimensiona...

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...tructed using the sheaf of formal microdifferential operators of order ≤ 0 of [22]. On a complex symplectic manifold, there is a canonical W0-algebroid. We refer to [21] for such constructions and to =-=[14]-=- for an application to the representation theory of rational Cherednik 42(4.27) algebras. In fact, the constructions of [21] are done using an important Csubalgebra of OX[[�]] consisting of sections ...

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...ry of such modules, introduced by Opdam and Rouquier [15], is denoted by O(Hc) and displays many similarities with “category O” for semisimple complex Lie algebras; this point of view is explained in =-=[19]-=-. The natural homomorphism C[h] → Hc allows us to think of such modules as coherent sheaves on the complex variety h. By completing at 0Date: May, 2010. 0 REPRESENTATIONS OF THE RATIONAL CHEREDNIK ALG...