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On matrix coefficients of the reflection representation

Authors: J. Matthew Douglass and Brad Shelton
Journal: Proc. Amer. Math. Soc. 105 (1989), 62-65
MSC: Primary 17B35; Secondary 20C30, 20G05
MathSciNet review: 930242
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Abstract: We answer in the affirmative a question of Lusztig on the signs of certain matrix coefficients for the reflection representation of the Hecke algebra of a finite Weyl group. In the action of the Weyl group on the associated root system, the coefficients of the action of an element on a root are all of the same sign. It is shown that an appropriate generalization of this property holds for the reflection representation.

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  • [1] A. Beilinson, Localization of representations of reductive Lie algebras, Proc. Internat. Cong. Math., North-Holland, 1983, pp. 699-710. MR 804725 (86m:17007)
  • [2] A. Beilinson and J. Bernstein, Localisation de $ \mathfrak{g}$-modulesi, C. R. Acad. Sci. Paris 292 (1981), 15-18. MR 610137 (82k:14015)
  • [3] J. L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. MR 632980 (83e:22020)
  • [4] D. Collingwood, R. Irving and B. Shelton, Filtrations on generalized Verma modules for Hermitian symmetric pairs, J. Reine Angew. Math. (to appear). MR 921987 (89a:22030)
  • [5] C. Curtis, N. Iwahori and R. Kilmoyer, Hecke algebras and characters of parabolic type of finite groups with $ (B,N)$-pairs, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 81-116. MR 0347996 (50:494)
  • [6] J. M. Douglass, Cells in Weyl groups corresponding to the reflection representation, preprint.
  • [7] O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. École Norm. Sup. 14 (1981), 261-302. MR 644519 (83e:17009)
  • [8] R. Irving, The socle filtration of a Verma module, preprint. MR 944101 (89h:17015)
  • [9] J. Jantzen, Moduln mit einem Hochsten Gewicht, Lecture Noteis in Math., vol. 750, Springer-Verlag, Berlin and New York, 1980. MR 552943 (81m:17011)
  • [10] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 560412 (81j:20066)
  • [11] G. Lusztig, Unipotent characters of the symplectic and odd orthogonal groups over a finite field, Invent. Math. 64 (1981), 263-296. MR 629472 (83b:20011)
  • [12] H. Tiwari, Reflection representations of Hecke algebras of certain Weyl groups, preprint. MR 977868 (89k:20072)

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Article copyright: © Copyright 1989 American Mathematical Society

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