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


SIGMA 5 (2009), 057, 18 pages      arXiv:0812.2624      https://doi.org/10.3842/SIGMA.2009.057
Contribution to the Special Issue on Dunkl Operators and Related Topics

Dunkl Operators and Canonical Invariants of Reflection Groups

Arkady Berenstein a and Yurii Burman b, c
a) Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
b) Independent University of Moscow, 11 B. Vlassievsky per., 121002 Moscow, Russia
c) Higher School of Economics, 20 Myasnitskaya Str., 101000 Moscow, Russia

Received December 14, 2008, in final form May 21, 2009; Published online June 03, 2009

Abstract
Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions.

Key words: Dunkl operators; reflection group.

pdf (330 kb)   ps (217 kb)   tex (24 kb)

References

  1. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and Its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  2. Berenstein A., Burman Yu., Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras, Trans. Amer. Math. Soc., to appear, math.RT/0505173.
  3. Berest Yu., The problem of lacunas and analysis on root systems, Trans. Amer. Math. Soc. 352 (2000), 3743-3776.
  4. Berest Y., Etingof P., Ginzburg V., Finite-dimensional representations of rational Cherednik algebras, Int. Math. Res. Not. 2003 (2003), no. 19, 1053-1088, math.RT/0208138.
  5. Broue M., Malle G., Rouquier R., Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127-190.
  6. Chevalley C., Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782.
  7. Cohen A., Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), 379-436.
  8. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  9. Dunkl C.F., Intertwining operators and polynomials associated with symmetric group, Monatsh. Math. 126 (1998), 181-209.
  10. Dunkl C.F., Polynomials associated with dihedral groups, SIGMA 3 (2007), 052, 19 pages, math.CA/0702107.
  11. Dunkl C.F, de Jeu M.F.E., Opdam E.M., Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237-256.
  12. Dunkl C.F., Xu Yu., Orthogonal polynomials of several variables, Cambridge University Press, Cambridge, 2001.
  13. Etingof P., Stoica E., Griffeth S., Unitary representations of rational Cherednik algebras, arXiv:0901.4595.
  14. Heckman G.J., A remark on the Dunkl differential-difference operators, in Harmonic analysis on reductive groups (Brunswick, ME, 1989), Progr. Math., Vol. 101, Birkhäuser Boston, Boston, MA, 1991, 181-191.
  15. Iwasaki K., Basic invariants of finite reflection groups, J. Algebra 195 (1997), 538-547.
  16. Kostant B., Clifford algebra analogue of the Hopf-Koszul-Samelson theorem, the ρ-decomposition C(g) = End Vρ Ä C(P), and the g-module structure of Ùg, Adv. Math. 125 (1997), 275-350.
  17. Lapointe L., Lascoux A., Morse J., Determinantal expression and recursion for Jack polynomials, Electron. J. Combin. 7 (2000), Note 1, 7 pages.
  18. Macdonald I., Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge, 2003.
  19. Opdam E., Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121.
  20. Steinberg R., Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392-400.


Previous article   Next article   Contents of Volume 5 (2009)