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QUANTUM SYMMETRIC PAIRS AND REPRESENTATIONS OF DOUBLE AFFINE HECKE ALGEBRAS OF TYPE (C ∨ n, Cn)
, 908
"... Abstract. We build representations of the affine and double affine braid groups and Hecke algebras of type (C ∨ n, Cn), based upon the theory of quantum symmetric pairs (U, B). In the case U = Uq(glN), our constructions provide a quantization of the representations constructed by Etingof, Freund and ..."
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Abstract. We build representations of the affine and double affine braid groups and Hecke algebras of type (C ∨ n, Cn), based upon the theory of quantum symmetric pairs (U, B). In the case U = Uq(glN), our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type BC generalization of the results in arXiv:0805.2766. 1.
Quantum Dmodules, elliptic braid groups, and double affine Hecke algebras
 IMRN 2009
"... Abstract. We build representations of the elliptic braid group from the data of a quantum Dmodule M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid [Ly], [LyMa], and also certain geometric constructions of Calaque, Enriq ..."
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Abstract. We build representations of the elliptic braid group from the data of a quantum Dmodule M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid [Ly], [LyMa], and also certain geometric constructions of Calaque, Enriquez, and Etingof [CEE] concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasiclassical limit of U = Ut(slN), as t → 1. In the latter case, we produce representations of the double affine Hecke algebra of type An−1, for each n.
Quantum dimensions and their nonArchimedean degenerations
 IMRP Int. Math. Res. Pap. (2006), Art. ID
"... Abstract. We derive explicit dimension formulas for irreducible MFspherical KFrepresentations where KF is the maximal compact subgroup of the general linear group GLd(F) over a local field F and MF is a closed subgroup of KF such that KF/MF realizes the Grassmannian of ndimensional Fsubspaces of ..."
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Abstract. We derive explicit dimension formulas for irreducible MFspherical KFrepresentations where KF is the maximal compact subgroup of the general linear group GLd(F) over a local field F and MF is a closed subgroup of KF such that KF/MF realizes the Grassmannian of ndimensional Fsubspaces of F d. We explore the fact that (KF, MF) is a Gelfand pair whose associated zonal spherical functions identify with various degenerations of the multivariable little qJacobi polynomials. As a result, we are led to consider generalized dimensions defined in terms of evaluations and quadratic norms of multivariable little qJacobi polynomials, which interpolate between the various classical dimensions. The generalized dimensions themselves are shown to have representation theoretic interpretations as the quantum dimensions of irreducible spherical quantum representations associated to quantum complex Grassmannians. 1.
THE CENTER OF QUANTUM SYMMETRIC PAIR COIDEAL SUBALGEBRAS
"... Abstract. The theory of quantum symmetric pairs as developed by the second author is based on coideal subalgebras of the quantized universal enveloping algebra for a semisimple Lie algebra. This paper investigates the center of these coideal subalgebras, proving that the center is a polynomial ring. ..."
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Abstract. The theory of quantum symmetric pairs as developed by the second author is based on coideal subalgebras of the quantized universal enveloping algebra for a semisimple Lie algebra. This paper investigates the center of these coideal subalgebras, proving that the center is a polynomial ring. A basis of the center is given in terms of a submonoid of the dominant integral weights.
The dynamical U(n) quantum group
 Int. J. Math. Math. Sci. (2006) Art. ID
"... Abstract. We study the dynamical analogue of the matrix algebra M(n), constructed from a dynamical Rmatrix given by Etingof and Varchenko. A left and a right corepresentation of this algebra, which can be seen as analogues of the exterior algebra representation, are defined and this defines dynamic ..."
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Abstract. We study the dynamical analogue of the matrix algebra M(n), constructed from a dynamical Rmatrix given by Etingof and Varchenko. A left and a right corepresentation of this algebra, which can be seen as analogues of the exterior algebra representation, are defined and this defines dynamical quantum minor determinants as the matrix elements of these corepresentations. These elements are studied in more detail, especially the action of the comultiplication and Laplace expansions. Using the Laplace expansions we can prove that the dynamical quantum determinant is almost central, and adjoining an inverse the antipode can be defined. This results in the dynamical GL(n) quantum group associated to the dynamical Rmatrix. We study a ∗structure leading to the dynamical U(n) quantum group, and we obtain results for the canonical pairing arising from the Rmatrix. 1.
PoissonLie interpretation of trigonometric Ruijsenaars duality
, 906
"... A geometric interpretation of the duality between two real forms of the complex trigonometric RuijsenaarsSchneider model is presented. The phase spaces of the models in duality are realized as two different gauge slices in the same inverse image of the moment map defining a suitable symplectic redu ..."
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A geometric interpretation of the duality between two real forms of the complex trigonometric RuijsenaarsSchneider model is presented. The phase spaces of the models in duality are realized as two different gauge slices in the same inverse image of the moment map defining a suitable symplectic reduction of the standard Heisenberg double of U(n). The collections of commuting Hamiltonians of the models in duality are shown to descend from two families of ‘free ’ Hamiltonians on the double which are dual to each other in a PoissonLie sense. Our results give rise to a major simplification of Ruijsenaars’ proof of the crucial symplectomorphism property of the duality map. 1 1
JACK POLYNOMIALS FOR THE BCn ROOT SYSTEM AND GENERALIZED SPHERICAL FUNCTIONS
, 2002
"... Functions on a homogeneous space G/K invariant with respect to the left action of K are called spherical functions (or sometimes Kspherical). One can also study functions on G/K with values in a representation V of G which are equivariant with respect to the left action of K. This more ..."
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Functions on a homogeneous space G/K invariant with respect to the left action of K are called spherical functions (or sometimes Kspherical). One can also study functions on G/K with values in a representation V of G which are equivariant with respect to the left action of K. This more