On the category 𝒪 for rational Cherednik algebras

By V. Ginzburg, N. Guay, E. Opdam, Raphaël Rouquier. 2002

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Abstract We study the category 𝒪 of representations of the rational Cherednik algebra AW attached to a complex reflection group W. We construct an exact functor, called Knizhnik-Zamolodchikov functor: 𝒪→ℋW-mod, where ℋW is the (finite) Iwahori-Hecke algebra associated to W. We prove that the Knizhnik-Zamolodchikov functor induces an equivalence between 𝒪/𝒪tor, the quotient of 𝒪 by the subcategory of AW-modules supported on the discriminant, and the category of finite-dimensional ℋW-modules. The standard AW-modules go, under this equivalence, to certain modules arising in Kazhdan-Lusztig theory of “cells”, provided W is a Weyl group and the Hecke algebra ℋW has equal parameters. We prove that the category 𝒪 is equivalent to the module category over a finite dimensional algebra, a generalized “q-Schur algebra” associated to W.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2c6a0daf960f6705f76724f886082e346aa07917

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Citations in the corpus

Categorification of (induced) cell modules and the rough structure of generalized Verma modules (2007, unknown): “This filtration is based on the Gelfand–Kirillov dimension which is closely related to Lusztig’s a-function (see [50]) and should be compared with the appearance of the a-function in [12] and [28].”
Finite-dimensional representations of rational Cherednik algebras (2002, background): “The proof in [16] is much more difficult and involves Kazhdan-Lusztig theory of cells.”
Catalan numbers for complex reflection groups (2009, methodology): “The ﬁrst is even an equivalence of categories, [4, Corol-lary 2.20 and Theorem 5.14], and so gives a bijection 2.6.”
q-Schur algebras and complex reflection groups, I (2005, methodology): “ory O of rational Cherednik algebras : we show that, in type A, when the parameter is not in 1 2 + Z, the category O is equivalent to the module category of a q-Schur algebra, solving a conjecture of [GGOR]. As a consequence, we obtain character formulas for simple objects of O in this case. We also”
Projective-injective modules, Serre functors and symmetric algebras (2005, background): “The naive duality ([31], Proposition 4.”
Finite dimensional representations of DAHA and affine Springers fibers : the spherical case (2007, unknown): “w→w, ai→−2ˇai/(ˇai,ˇai), ˇai→2ai/(ai,ai), κ→κ. The subalgebras generated by V0,Vˇ0 are isomorphic to the polynomial algebras Cκ[Vˇ0], Cκ[V0] respectively. Recall the subcategory O(H′′) ⊂mod(H′′) from [GGOR], which consists of the ﬁnitely generated modules which are locally ﬁnite over Cκ[Vˇ0] and suc”
Rational Cherednik algebras and Hilbert schemes (2004, background): “nd µ∈Irrep(W) (2.8.1) KZ(∆c(µ)) ∼=Spq(µ)∗ where Spq(µ) is the so-called Specht module associated to µ. (The dual module appears since the deﬁning relations for the rational Cherednik algebra given in [GGOR] are normalized diﬀerently to (2.1); as remarked in (2.1), our parameter ccorresponds to −k1 i”
On category O for cyclotomic rational Cherednik algebras (2011, background): “This category has a highest weight structure where the standard modules 1(λ) and simple modules L(λ) are parameterized by irreducible W -modules λ [31].”
On the quotient ring by diagonal invariants (2002, methodology): “We will now outline the arguments of [15] and [16] which show that Oc is a highest weight catgeory in the sense of [6].”