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

Gordon, Iain

doi:10.1093/imrp/rpn006

Cited by 38 publications

(67 citation statements)

References 45 publications

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“…It seems likely that the orders described in Gordon's paper [Gor2] arise in this way for r > 1; if this is the case, then a positive answer to Question 10.1 of Gordon's paper should follow. For c 0 / ∈ Z + 1 2 , the next corollary follows from Rouquier's work [Rou] and the corresponding result for the q-Schur algebra.…”

Section: Lower Terms This Proves Part (A)mentioning

confidence: 98%

Orthogonal functions generalizing Jack polynomials

Griffeth

2010

Trans. Amer. Math. Soc.

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Abstract. The rational Cherednik algebra H is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M (λ) for H. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible W -modules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M (λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M (λ) consisting of orthogonal functions on C n with values in the representation S λ . For G(1, 1, n) with λ = (n) these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M (λ) in the case in which the orthogonal functions are all well-defined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of H so that the rational Cherednik algebra for G (r, p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G (r, p, n) by Clifford theory.

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Section: Lower Terms This Proves Part (A)mentioning

confidence: 98%

Orthogonal functions generalizing Jack polynomials

Griffeth

2010

Trans. Amer. Math. Soc.

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“…[16,Section 7.2.3] or [13,Lemma 7.8]) that M 0 (W, V ) ∼ = A 2n /Γ n where Γ n is the wreath product Z/rZ S n , where n depends on ν as follows. Let δ stand for the dimension vector (1, .…”

Section: 1mentioning

confidence: 99%

“…We define L μ as the closure of L • μ . I. Gordon [13,Section 5.4] has defined a partial order on P(r, n) generated by the rule μ…”

Section: Lagrangian Componentsmentioning

confidence: 99%

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Wreath Macdonald polynomials and the categorical McKay correspondence

Bezrukavnikov

Finkelberg

Vologodsky

2014

Cambridge Journal of Mathematics

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with an Appendix by Vadim Vologodsky To the memory of Andrei ZelevinskyMark Haiman has reduced Macdonald Positivity Conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product S n (Z/rZ) n . He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of A 2n by the symmetric group S n .A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin [2] via quantization in positive characteristic. In the present note we prove various properties of these derived equivalences and then deduce generalized Macdonald positivity for wreath products.

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“…A similar algebraic analysis is carried out for kleinian singularities, [12] and [57], and for Cherednik algebras with W = G(d, 1, n), [41], but in this general case the geometry of the associated varieties generalising Hilb n (C 2 ) is not yet completely understood. There is also a localisation theorem for Harish-Chandra bimodules of finite W -algebras in this spirit, [36].…”

Section: Remarksmentioning

confidence: 99%