On the category 𝒪 for rational Cherednik algebras

Ginzburg, Victor; Guay, Nicolas; Opdam, Eric; Rouquier, Raphaël

doi:10.1007/s00222-003-0313-8

Cited by 212 publications

(387 citation statements)

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“…Confirming (18). Suppose that we have a charge s and a partition λ, so that the corresponding β-numbers are x j = λ j + s + 1 − j for j ≥ 1.…”

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Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

Gordon¹

2010

International Mathematics Research Papers

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Abstract. We relate the representations of the rational Cherednik algebras associated with the complex reflection group µ ℓ ≀ Sn to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Z-algebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change.This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function (the c-function) used to define a highest weight structure on category O of the Cherednik algebra.This interpretation provides a natural partial ordering on O which we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step towards a geometric picture for that.We also interpret geometrically another ordering function (the a-function) used in the study of Hecke algebras. (The connection between Cherednik algebras and Hecke algebras is provided by the KZ-functor.) This is related to a conjecture of Bonnafé and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should (and do for type B) correspond to the G.I.T. chambers defining the quiver varieties. As a result anything that can be defined via the quiver varieties, including the a-function, will be constant on these classes.

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“…Confirming (18). Suppose that we have a charge s and a partition λ, so that the corresponding β-numbers are x j = λ j + s + 1 − j for j ≥ 1.…”

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confidence: 87%

“…We prove first that (18) n( t λ (r) ) − n(λ (r) ) and then that (19) (p,q)∈τs (λ) ǫ cont(p,q) cont(p, q) .…”

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confidence: 96%

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

Gordon¹

2010

International Mathematics Research Papers

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“…Inspired by the papers [Ginzburg et al 2003;Guay 2005], we suggest a notion of category ᏻ for the Lie algebra sl n (H t=1,c ( )), and we study certain modules in it, which we call quasifinite highest weight modules.…”

Section: Highest Weight Representations For Matrix Lie Algebras Over mentioning

confidence: 99%

Double affine Lie algebras and finite groups

Guay¹,

Hernandez²,

Loktev³

2009

Pacific J. Math.

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We begin to study the Lie theoretical analogues of symplectic reflection algebras for a finite cyclic group ; we call these algebras "cyclic double affine Lie algebras". We focus on type A: In the finite (respectively affine, double affine) case, we prove that these structures are finite (respectively affine, toroidal) type Lie algebras, but the gradings differ. The case that is essentially new is sl n (‫ [ރ‬u, v] ). We describe its universal central extensions and start the study of its representation theory, in particular of its highest weight integrable modules and Weyl modules. We also consider the first Weyl algebra A 1 instead of the polynomial ring ‫ [ރ‬u, v], and, more generally, a rank one rational Cherednik algebra. We study quasifinite highest weight representations of these Lie algebras.

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“…This technique has been extended by Geck and Rouquier [14] to a decent class of algebras over integral domains. It became an important tool for studying algebras involving parameters, so for example Hecke algebras (see [13], [12], and [7]) and, more recently, rational Cherednik algebras (see [2], [15], and [28]). We list several further examples in §2A.…”

Section: Introductionmentioning

confidence: 99%

Decomposition Matrices are Generically Trivial

Thiel¹

2015

Int Math Res Notices

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ABSTRACT. We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.

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On the category 𝒪 for rational Cherednik algebras

Cited by 212 publications

References 20 publications

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

Double affine Lie algebras and finite groups

Decomposition Matrices are Generically Trivial

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