Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 055, 11 pages      arXiv:1507.08365      https://doi.org/10.3842/SIGMA.2016.055

A Family of Finite-Dimensional Representations of Generalized Double Affine Hecke Algebras of Higher Rank

Yuchen Fu and Seth Shelley-Abrahamson
Department of Mathematics, Massachusetts Institute of Technology, 182 Memorial Drive, Cambridge, MA 02139, USA

Received April 20, 2016, in final form June 11, 2016; Published online June 14, 2016

Abstract
We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using $R$-matrices for $U_q(\mathfrak{sl}_N)$. Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.

Key words: generalized double affine Hecke algebra; $R$-matrix; Drinfeld-Kohno theorem.

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References

  1. Arakawa T., Suzuki T., Duality between ${\mathfrak{sl}}_n({\mathbb C})$ and the degenerate affine Hecke algebra, J. Algebra 209 (1998), 288-304.
  2. Calaque D., Enriquez B., Etingof P., Universal KZB equations: the elliptic case, in Algebra, Arithmetic, and Geometry: in Honor of Yu.I. Manin, Vol. I, Progr. Math., Vol. 269, Birkhäuser Boston, Inc., Boston, MA, 2009, 165-266, math.QA/0702670.
  3. Drinfeld V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1989), 1419-1457.
  4. Etingof P., Gan W.L., Oblomkov A., Generalized double affine Hecke algebras of higher rank, J. Reine Angew. Math. 600 (2006), 177-201, math.QA/0504089.
  5. Etingof P., Oblomkov A., Rains E., Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces, Adv. Math. 212 (2007), 749-796, math.QA/0406480.
  6. Etingof P.I., Frenkel I.B., Kirillov Jr. A.A., Lectures on representation theory and Knizhnik-Zamolodchikov equations, Mathematical Surveys and Monographs, Vol. 58, Amer. Math. Soc., Providence, RI, 1998.
  7. Fulton W., Harris J., Representation theory. A first course, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.
  8. Jantzen J.C., Lectures on quantum groups, Graduate Studies in Mathematics, Vol. 6, Amer. Math. Soc., Providence, RI, 1996.
  9. Jordan D., Quantum $D$-modules, elliptic braid groups, and double affine Hecke algebras, Int. Math. Res. Not. 2009 (2009), 2081-2105, arXiv:0805.2766.
  10. Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
  11. Kazhdan D., Lusztig G., Affine Lie algebras and quantum groups, Int. Math. Res. Not. 1991 (1991), 21-29.
  12. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  13. Kohno T., Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier (Grenoble) 37 (1987), 139-160.
  14. Montarani S., Representations of Gan-Ginzburg algebras, Selecta Math. (N.S.) 16 (2010), 631-671, arXiv:1001.2588.
  15. Orellana R., Ram A., Affine braids, Markov traces and the category ${\mathcal O}$, in Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007, 423-473, math.RT/0401317.

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