Unitary Hecke algebra modules with nonzero Dirac cohomology

  • Dan BarbaschEmail author
  • Dan Ciubotaru
Part of the Progress in Mathematics book series (PM, volume 257)


In this paper, we review the construction of the Dirac operator for graded affine Hecke algebras and calculate the Dirac cohomology of irreducible unitary modules for the graded Hecke algebra of gl(n).


Dirac cohomology Unitary representations Hecke algebra 

Mathematics Subject Classification

22 16 20 


  1. [A]
    J. Arthur, Unipotent automorphic representations: conjectures. Orbites unipotentes et représentations, II, Astérisque No. 171–172 (1989), 13–71.Google Scholar
  2. [AS]
    M. Atiyah, W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62.CrossRefMathSciNetzbMATHGoogle Scholar
  3. [BCT]
    D. Barbasch, D. Ciubotaru, P. Trapa, The Dirac operator for graded affine Hecke algebras, Acta Math. 209 (2012), 197–227.CrossRefMathSciNetzbMATHGoogle Scholar
  4. [BM]
    D. Barbasch, A. Moy, Unitary spherical spectrum for p-adic classical groups, Acta Appl. Math. 44 (1996), no. 1–2, 3–37.CrossRefMathSciNetzbMATHGoogle Scholar
  5. [BMcP]
    W. Borho, R. MacPherson, Partial resolutions of nilpotent varieties, Analysis and topology on singular spaces, II, III (Luminy, 1981), 23–74, Astérisque, 101–102, Soc. Math. France, Paris, 1983.Google Scholar
  6. [BZ]
    J. Bernstein, A. Zelevinsky, Induced representations of reductive p-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472.Google Scholar
  7. [BW]
    A. Borel, N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Princeton University Press, Princeton, New Jersey, 1980.zbMATHGoogle Scholar
  8. [Ch]
    C. Chevalley, The Algebraic Theory of Spinors, Columbia University Press, New York, 1954. viii+131 pp.Google Scholar
  9. [Ci]
    D. Ciubotaru, Spin representations of Weyl groups and Springer’s correspondence, J. Reine Angew. Math. 671 (2012), 199–222.MathSciNetzbMATHGoogle Scholar
  10. [CT]
    D. Ciubotaru, P. Trapa, Characters of Springer representations on elliptic conjugacy classes, Duke Math. J. 162 (2013), 201–223.CrossRefMathSciNetzbMATHGoogle Scholar
  11. [EW]
    T. Enright, N. Wallach, Embeddings of unitary highest weight representations and generalized Dirac operators, Math. Ann. 307 (1997), no. 4, 627–646.CrossRefMathSciNetzbMATHGoogle Scholar
  12. [HP]
    J.-S. Huang, P. Pandzić, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), no. 1, 185–202.CrossRefMathSciNetzbMATHGoogle Scholar
  13. [KZ]
    A. Knapp, G. Zuckerman, Classification theorems for representations of semisimple Lie groups, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), 138–159. Lecture Notes in Math., Vol. 587, Springer, Berlin, 1977.Google Scholar
  14. [Lu]
    G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599–635.CrossRefMathSciNetzbMATHGoogle Scholar
  15. [Ma]
    I.G. MacDonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp.Google Scholar
  16. [Mo]
    A. Morris, Projective representations of reflection groups. II, Proc. London Math. Soc. (3) 40 (1980), no. 3, 553–576.Google Scholar
  17. [OV]
    A. Okounkov, A. Vershik, A new approach to representation theory of symmetric groups, Selecta Math. 2 (4) (1996), 581–605.CrossRefMathSciNetzbMATHGoogle Scholar
  18. [P]
    R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30.Google Scholar
  19. [Ro]
    J. Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math. 79 (1985), no. 3, 443–465.CrossRefMathSciNetzbMATHGoogle Scholar
  20. [St]
    J. Stembridge, Shifted tableaux and the projective representations of symmetric groups, Adv. Math. 74 (1989), no. 1, 87–134.CrossRefMathSciNetzbMATHGoogle Scholar
  21. [Ta]
    M. Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335–382.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Mathematics InstituteUniversity of OxfordOxfordEngland

Personalised recommendations