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


SIGMA 5 (2009), 012, 27 pages      arXiv:0810.2068      http://dx.doi.org/10.3842/SIGMA.2009.012
Contribution to the Special Issue on Dunkl Operators and Related Topics

Hecke-Clifford Algebras and Spin Hecke Algebras IV: Odd Double Affine Type

Ta Khongsap and Weiqiang Wang
Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA

Received October 15, 2008, in final form January 22, 2009; Published online January 28, 2009

Abstract
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.

Key words: spin Hecke algebras; Hecke-Clifford algebras; Dunkl operators.

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References

  1. Bazlov Y., Berenstein A., Noncommutative Dunkl operators and braided Cherednik algebras, arXiv:0806.0867.
  2. Drinfeld V., Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 58-60.
  3. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  4. Dunkl C.F., Opdam E.M., Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70-108, math.RT/0108185.
  5. Etingof P., Ginzburg V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348, math.AG/0011114.
  6. Ihara S., Yokonuma T., On the second cohomology groups (Schur-multipliers) of finite reflection groups, J. Fac. Sci. Univ. Tokyo Sect. I 11 (1965), 155-171.
  7. Karpilovsky G., The Schur multiplier, London Mathematical Society Monographs, New Series, Vol. 2, The Clarendon Press, Oxford University Press, New York, 1987.
  8. Khongsap T., Hecke-Clifford algebras and spin Hecke algebras III: the trigonometric type, arXiv:0808.2951.
  9. Khongsap T., Wang W., Hecke-Clifford algebras and spin Hecke algebras I: the classical affine type, Transform. Groups 13 (2008), 389-412, arXiv:0704.0201.
  10. Khongsap T., Wang W., Hecke-Clifford algebras and spin Hecke algebras II: the rational double affine type, Pacific J. Math. 238 (2008), 73-103, arXiv:0710.5877.
  11. Lusztig G., Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635.
  12. Morris A., Projective representations of reflection groups, Proc. London Math. Soc. (3) 32 (1976), 403-420.
  13. Nazarov M., Young's symmetrizers for projective representations of the symmetric group, Adv. Math. 127 (1997), 190-257.
  14. Rouquier R., Representations of rational Cherednik algebras, in Infinite-Dimensional Aspects of Representation Theory and Applications (Charlottesville, 2004), Contemp. Math. 392 (2005), 103-131, math.RT/0504600.
  15. Schur I., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250.
  16. Wang W., Double affine Hecke algebras for the spin symmetric group, math.RT/0608074.
  17. Wang W., Spin Hecke algebras of finite and affine types, Adv. Math. 212 (2007), 723-748, math.RT/0611950.

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