Rational Cherednik algebras and Hilbert schemes

Gordon, Iain; Stafford, J. T.

doi:10.1016/j.aim.2004.12.005

Cited by 71 publications

(155 citation statements)

References 36 publications

Supporting

Mentioning

150

Contrasting

Order By: Relevance

“…[22,Section 3] for another construction of the shift functors; however, the main result of [17] shows that the functors of [22] agree with the definition given here.…”

Section: Chamber Decompositionsmentioning

confidence: 81%

“…To relate these varieties with Cherednik algebras we recall that when ℓ = 1 H h provides a quantisation of the Hilbert scheme of n points on the plane, the relevant quiver variety in this special case, [22]. This quantisation is constructed by showing that the Opdam-Heckman shift functors for H h are noncommutative analogues of powers of an ample line bundle that appears naturally in the quiver theoretic description of the Hilbert scheme.…”

Section: Quiver Varietiesmentioning

confidence: 99%

“…We would like to relate the representation theory of H 1,h with the geometry of the spaces M θ (n). To do this we combine the Z-algebra formalism of [22] with the differential operator approach of [20]. We will use the definitions and notation of Z-algebras from [22] without further comment: the interested reader should consult that paper for details, in particular Section 5 there.…”

Section: Chamber Decompositionsmentioning

confidence: 99%

See 2 more Smart Citations

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

Gordon¹

2010

International Mathematics Research Papers

View full text Add to dashboard Cite

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.

show abstract

“…[22,Section 3] for another construction of the shift functors; however, the main result of [17] shows that the functors of [22] agree with the definition given here.…”

Section: Chamber Decompositionsmentioning

confidence: 81%

Section: Quiver Varietiesmentioning

confidence: 99%

Section: Chamber Decompositionsmentioning

confidence: 99%

See 1 more Smart Citation

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

Gordon¹

2010

International Mathematics Research Papers

View full text Add to dashboard Cite

show abstract

“…One has an isomorphism gr( κ+1 Q κ ) = A, see [GGS,Theorem 5.3] and also [GS1,Lemma 6.9(2)]. We deduce that the map gr…”

Section: ]mentioning

confidence: 94%

Hamiltonian reduction and nearby cycles for mirabolic D-modules

Bellamy

Ginzburg

2015

Advances in Mathematics

View full text Add to dashboard Cite

ABSTRACT. We study holonomic D-modules on SLn(C) × C n , called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus.We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O, with the restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic Dmodules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

show abstract

“…The noetherian property is immediate from Proposition 5.2.1, and the isomorphism gr K ( , ) ∼ = ♯ A ( ) ≽ follows from Proposition 5.2.1(iii). The equivalence statement in the theorem is a consequence of properties (iii) and (iv) above, thanks to Gordon and Stafford [GS,Lemma 5.5], and Boyarchenko [Bo,Theorem 4.4]. □ Remark 6.3.3.…”

Section: Victor Ginzburgmentioning

confidence: 99%

Harish-Chandra bimodules for quantized Slodowy slices

Ginzburg

2009

Represent. Theory

View full text Add to dashboard Cite

Abstract. The Slodowy slice is an especially nice slice to a given nilpotent conjugacy class in a semisimple Lie algebra. Premet introduced noncommutative quantizations of the Poisson algebra of polynomial functions on the Slodowy slice.In this paper, we define and study Harish-Chandra bimodules over Premet's algebras. We apply the technique of Harish-Chandra bimodules to prove a conjecture of Premet concerning primitive ideals, to define projective functors, and to construct "noncommutative resolutions" of Slodowy slices via translation functors.

show abstract

Rational Cherednik algebras and Hilbert schemes

Cited by 71 publications

References 36 publications

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

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

Hamiltonian reduction and nearby cycles for mirabolic D-modules

Harish-Chandra bimodules for quantized Slodowy slices

Contact Info

Product

Resources

About