Generalized double affine Hecke algebras of higher rank

Etingof, Pavel; Gan, Wee Liang; Oblomkov, Alexei

doi:10.1515/crelle.2006.091

Cited by 14 publications

(29 citation statements)

References 9 publications

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“…Our main result is a quantum analog of the statement from classical symplectic geometry, stating that this wreath product Calogero-Moser space may be obtained by classical Hamiltonian reduction from a product of m coadjoint orbits of the Lie algebra sl n , or, equivalently, as the space of solutions of a special kind of the additive Deligne-Simpson problem. This classical result can be found in [EGO, Sect. 2.6].…”

mentioning

confidence: 54%

A Lie-Theoretic Construction of Spherical Symplectic Reflection Algebras

Etingof

Loktev

Oblomkov

et al. 2008

Transformation Groups

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We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semisimple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra. IntroductionThe main result of this paper is the realization of the spherical subalgebra of the wreath product symplectic reflection algebra of rank n of types D 4 , E 6 , E 7 , E 8 , introduced in [EG], as a quantum Hamiltonian reduction of the tensor product of m quotients of the enveloping algebra U (sl n ), where is 2, 3, 4, and 6, and m is 4, 3, 3, and 3, respectively. This allows one to define a functor which attaches a representation of the spherical symplectic reflection algebra to a collection of m representations of sl n , which are annihilated by certain ideals. In particular, this gives an explicit Lie-theoretic construction of many finite-dimensional representations of spherical symplectic reflection algebras, most of which appear to be new. In the rank 1 case, all finite-dimensional representations of spherical symplectic reflection algebras are classified by Crawley-Boevey and Holland [CBH] and our construction yields several explicit Lie-theoretic realizations of all of them.The proof of the main result is based on the previous work [EGGO], in which the spherical subalgebra of a wreath product symplectic reflection algebra (of any type) is realized as the quantum Hamiltonian reduction from the algebra of differential operators on representations (with a certain dimension vector) of the Calogero-Moser quiver, obtained from the corresponding extended Dynkin quiver by adjoining an auxiliary vertex, linked to the extending vertex. Namely, in the case when the extended Dynkin graph is star-shaped (which happens in the cases D 4 , E 6 , E 7 , E 8 ), this reduction can be performed in two steps: first the reduction with respect to the groups of basis changes at the noncentral vertices, and then with respect to the group of basis changes at the central (branching) vertex of the star. By the localization theorem for partial flag varieties, after the first step we obtain a tensor product of m quotients of enveloping algebras (the factors correspond to the branches of the extended Dynkin diagram), which yields the result.Recall that by the results of [EG], a wreath product spherical symplectic reflection algebra can be viewed as a quantization of the wreath product CalogeroMoser space, which is a deformation of the nth Hilbert scheme of the resolution of a Kleinian singularity. Our main result is a quantum analog of the statement from classical symplectic geometry, stating that this wreath product Calogero-Moser space may be obtained by classical Hamiltonian reduction from a product of m coadjoint orbits of the Lie algebra sl n , or, equivalently, as the space of solutions of a special kind of ...

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mentioning

confidence: 54%

A Lie-Theoretic Construction of Spherical Symplectic Reflection Algebras

Etingof

Loktev

Oblomkov

et al. 2008

Transformation Groups

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“…Results in [3] then imply that H is also closely related to several other algebras appearing in the literature, e.g. H is the spherical subalgebra of a certain deformed preprojective algebra by Gan and Ginzburg [7].…”

Section: Introductionmentioning

confidence: 74%

“…The algebra H has appeared earlier in the literature; it is isomorphic to a rational generalized DAHA defined by Etingof et al [3]. Results in [3] then imply that H is also closely related to several other algebras appearing in the literature, e.g.…”

Section: Introductionmentioning

confidence: 88%

“…We show in Sect. 3.4 that the algebra H(t) is isomorphic to the rational generalized double affine Hecke algebra attached to an affine Dynkin diagram of type D 4 as defined by Etingof et al [3].…”

Section: The Rational Double Affine Hecke Algebramentioning

confidence: 99%

“…We recall the definition here. Actually, in [3] a rational GDAHA of rank n is associated to any star-like graph that is not a finite Dynkin diagram.…”

Section: The Rational Gdaha For Type Dmentioning

confidence: 99%

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Multivariable Wilson polynomials and degenerate Hecke algebras

Groenevelt

2009

Sel. Math. New Ser.

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Abstract. We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type (C ∨ n , Cn). A certain representation in terms of difference-reflection operators naturally leads to the definition of nonsymmetric versions of the multivariable Wilson polynomials. Using the degenerate Hecke algebra we derive several properties, such as orthogonality relations and quadratic norms, for the nonsymmetric and symmetric multivariable Wilson polynomials. Mathematics Subject Classification (2000). Primary 33C52, 33C80; Secondary 20C08.

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Inozemtsev system as Seiberg-Witten integrable system

Argyres

Chalykh

Lü

2021

J. High Energ. Phys.

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In this work we establish that the Inozemtsev system is the Seiberg-Witten integrable system encoding the Coulomb branch physics of 4d $$ \mathcal{N} $$ N = 2 USp(2N) gauge theory with four fundamental and (for N ≥ 2) one antisymmetric tensor hypermultiplets. We describe the transformation from the spectral curves and canonical one-forms of the Inozemtsev system in the N = 1 and N = 2 cases to the Seiberg-Witten curves and differentials explicitly, along with the explicit matching of the modulus of the elliptic curve of spectral parameters to the gauge coupling of the field theory, and of the couplings of the Inozemtsev system to the field theory mass parameters. This result is a particular instance of a more general correspondence between crystallographic elliptic Calogero-Moser systems with Seiberg-Witten integrable systems, which will be explored in future work.

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Generalized double affine Hecke algebras of higher rank

Cited by 14 publications

References 9 publications

A Lie-Theoretic Construction of Spherical Symplectic Reflection Algebras

A Lie-Theoretic Construction of Spherical Symplectic Reflection Algebras

Multivariable Wilson polynomials and degenerate Hecke algebras

Inozemtsev system as Seiberg-Witten integrable system

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