Baby Verma Modules for Rational Cherednik Algebras

Gordon, Iain

doi:10.1112/s0024609303001978

Cited by 92 publications

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“…The fibre Υ −1 (0) is described in [19,Section 5]. In particular, its (closed) points are labelled by the isomorphism classes of simple G-modules, i.e.…”

Section: The C-function Topologicallymentioning

confidence: 99%

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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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“…The fibre Υ −1 (0) is described in [19,Section 5]. In particular, its (closed) points are labelled by the isomorphism classes of simple G-modules, i.e.…”

Section: The C-function Topologicallymentioning

confidence: 99%

“…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) .…”

mentioning

confidence: 99%

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

Gordon¹

2010

International Mathematics Research Papers

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“…to which is invariant under the action of so that descends to a function Ref( )/ → . To any such function the restricted rational Cherednik algebra H is defined (see [17]). This is a finite-dimensional -algebra obtained as a quotient of the correspond rational Cherednik algebra at = 0 introduced by Etingof and Ginzburg [10].…”

Section: Proposition Suppose Thatmentioning

confidence: 99%

“…These two cases are quite restrictive for certain applications, however. Restricted rational Cherednik algebras (see [17], [2], and [28]) for example are not covered by these results as in general neither the base rings are one-dimensional nor the generic fiber is semisimple (even though it is symmetric). Moreover, even results covered by the setting of Schur elements have the problem that we cannot apply them to restrictions /p for prime ideals p in general (what we would like to do to continue studying the fibers of on the locus where d p is non-trivial):…”

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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“…(2) In the case of the Calogero-Moser space, let Θ : Irr(W ) → X c (W ) be the morphism defined by [18], more details are in the body of the paper. The CM cells (or families) are, by definition, the fibers of this map.…”

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confidence: 99%

One-W-type Modules for Rational Cherednik Algebra and Cuspidal Two-sided Cells

Ciubotaru¹

2018

Bulletin of the Inst. of Math. Academia Sinica(N.S.)

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We classify the simple modules for the rational Cherednik algebra H0,c that are irreducible when restricted to W , in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in the sense of Lusztig. We compute the Dirac cohomology of these modules and use the tools of Dirac theory to find nontrivial relations between the cuspidal Calogero-Moser cells, in the sense of Bellamy, and the cuspidal two-sided cells.

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Baby Verma Modules for Rational Cherednik Algebras

Cited by 92 publications

References 16 publications

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

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

Decomposition Matrices are Generically Trivial

One-W-type Modules for Rational Cherednik Algebra and Cuspidal Two-sided Cells

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