Algebras and Representation Theory

, Volume 21, Issue 4, pp 859–895 | Cite as

Some Methods of Computing First Extensions Between Modules of Graded Hecke Algebras

  • Kei Yuen ChanEmail author


In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen filtrations. Our approach involves the use of information from the Langlands classification as well as some deeper understanding on some structure of some modules. Such module arises from the image of a Knapp-Stein type intertwining operator and is a quotient of a generalized standard module. As an application, we compute the Ext-groups for irreducible modules in a block for the graded Hecke algebra of type C 3, assuming the truth of a version of Jantzen conjecture.


Graded Hecke algebras Extensions Standard modules Filtrations 

Mathematics Subject Classification (2010)

20C08 16E30 22E50 


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The author would like to thank Dan Ciubotaru and Peter Trapa for discussions on Jantzen filtrations and would like to thank Eric Opdam for discussions on intertwining operators. The author would like to thank the referee for useful comments. This research was supported by both of the ERC-advanced grant no. 268105 from Eric Opdam and the Croucher Fellowship.


  1. 1.
    Adams, J., van Leeuwen, M., Trapa, P., Vogan, D.: Unitary representations of real reductive groups. arXiv:1212.2192
  2. 2.
    Arakawa, T., Suzuki, T.: Duality between \(\mathfrak {sl}(n,\mathbb {C})\) and the degenerate affine Hecke algebra. J. Algebra 209, 288–304 (1998)Google Scholar
  3. 3.
    Barbasch, D.: Filtrations on Verma modules. Ann. Sci. l’É,cole Norm. Sup. (4) 16, 489–494 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barbasch, D., Ciubotaru, D.: Hermitian forms for affine Hecke algebras. arXiv:1312.3316v1 [math.RT] (2015)
  5. 5.
    Barbasch, D., Ciubotaru, D.: Star operations for affine Hecke algebras. arXiv:1504.04361 [math.RT] (2015)
  6. 6.
    Barbasch, D., Moy, A.: A unitarity criterion for p-adic groups. Invent. Math. 98, 19–37 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barbasch, D., Moy, A.: Unitary spherical spectrum for p-adic classical groups. Acta Appl. Math. 44(1-2), 3–37 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beilinson, A., Bernstein, I.N.: A proof of Jantzen conjecture. Adv. Soviet Math. Part 1 16, 1–50 (1993)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Benson, D.J.: Representations and Cohomology. I: Basic Representation Theory of Finite Groups and Associative Algebras, 2nd Ed. Cambridge Studies in Advanced Mathematics, vol. 30. Cambridge University Press, Cambridge (1998)Google Scholar
  10. 10.
    Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, vol. 94. Princeton University Press, Princeton (1980)zbMATHGoogle Scholar
  11. 11.
    Chan, K.Y.: Extensions of Graded Affine Hecke Algebra Modules. PhD thesis, University of Utah (2014)Google Scholar
  12. 12.
    Chan, K.Y.: Duality for Ext-groups and extensions of discrete series for graded Hecke algebras. Adv. Math. 294, 410–453 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (1997)zbMATHGoogle Scholar
  14. 14.
    Ciubotaru, D.: Multiplicity matrices for the graded affine Hecke algebra. J. Algebra 320, 3950–3983 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Delorme, P., Opdam, E.M.: Analytic R-groups for affine Hecke algebras. J Reine Angew. Math. 658, 133–172 (2011)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Evens, S.: The Langlands classification for graded Hecke algebras. Proc. Amer. Math. Soc. 124(4), 1285–1290 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Evens, S., Mirković, I.: Fourier transform and the Iwahori-Matsumoto involution. Duke Math. J. 86(3), 435–464 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Humphreys, J.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  19. 19.
    Irving, R.S.: Projective modules in the category o S. Loewy series. Trans. Amer. Math. Soc. 291, 733–754 (1985)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Jantzen, J.C.: Modulen Mit Einem Höchsten Gewicht, Lect Notes in Math, vol. 750. Springer, Berlin (1979)CrossRefGoogle Scholar
  21. 21.
    Kato, S.: A homological study of Green polynomials, to appear in Ann. Sci. l’É,cole Norm. Sup. arXiv:1111.4640
  22. 22.
    Knapp, A.W.: Representation theory of semisimple Lie groups: an overview based on examples. Princeton University Press, Princeton (1986)CrossRefzbMATHGoogle Scholar
  23. 23.
    Knapp, A.W., Stein, E.M.: Intertwining operators for semisimple groups II. Invent. Math. 60, 9–84 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kriloff, C., Ram, A.: Representations of graded Hecke algebras. Represent. Theory 6, 31–69 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Langlands, R.: On the classification of irreducible representations of real algebraic groups. In: Representation Theory and Harmonic Analysis on Semisimple Lie Groups, 101-170, Math. Surveys Monogr., 31, Amer. Math. Soc., Providence, RI (1989)Google Scholar
  26. 26.
    Lusztig, G.: Cuspidal local systems and graded Hecke algebras, I. Publ. Math. IHÉ,S 67, 145–202 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lusztig, G.: Affine Hecke algebras and their graded versions. J. Amer. Math. Soc. 2, 599–635 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lusztig, G.: Study of perverse sheaves arising from graded Lie algebras. Adv. Math. 112, 147–217 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lusztig, G.: Graded Lie algebras and intersection cohomology, representation theory of algebraic groups and quantum groups. Progr. Math. 284, 191–244 (2010). Birkhäuser/Springer, New YorkzbMATHGoogle Scholar
  30. 30.
    Meyer, R.: Homological Algebra for Schwartz Algebras of Reductive p-Adic Groups, Noncommutative Geometry and Number Theory, Aspects of Mathematics E37, pp. 263–300. Vieweg Verlag, Wiesbaden (2006)Google Scholar
  31. 31.
    Opdam, E.M.: Harmonic analysis for certain representations of graded Hecke algebras. Acta Math. 175(1), 75–121 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Opdam, E.M., Solleveld, M.: Homological algebra for affine Hecke algebras. Adv. Math. 220, 1549–1601 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Opdam, E.M., Solleveld, M.: Extensions of tempered representations. GAFA, Geom. Funct. Anal. 23, 664–714 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Reeder, M.: Nonstandard intertwining operators and the structure of unramified principal series representations. Forum Math. 9, 457–516 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rogawski, J.: On modules over the Hecke algebra of a p-adic group. Invent. Math. 79, 443–465 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Suzuki, T.: Rogawski’s conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A. Represent. Theory 2, 393–409 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Solleveld, M.: Parabolically induced representations of graded Hecke algebras. Algebr. Represent. Theory 152, 233–271 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Vogan, D.: Irreducible characters of semisimple Lie groups. II. The Kazhdan-Lusztig conjectures. Duke Math. J. 46(4), 805–859 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Weibel, C.: An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA

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