Nagoya Mathematical Journal

Lusztig's $a$-function in type $B_{n}$ in the asymptotic case

Meinolf Geck and Lacrimioara Iancu

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Abstract

In this paper, we study Lusztig's ${\mathbf{a}}$-function for a Coxeter group with unequal parameters. We determine that function explicitly in the "asymptotic case" in type $B_{n}$, where the left cells have been determined in terms of a generalized Robinson-Schensted correspondence by Bonnafé and the second author. As a consequence, we can show that all of Lusztig's conjectural properties (P1)--(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the "leading matrix coefficients" introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori-Hecke algebra ${\mathcal{H}}_{n}$ in terms of the Dipper-James-Murphy basis of ${\mathcal{H}}_{n}$.

Article information

Source
Nagoya Math. J., Volume 182 (2006), 199-240.

Dates
First available in Project Euclid: 20 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1150810007

Mathematical Reviews number (MathSciNet)
MR2235342

Zentralblatt MATH identifier
1173.20301

Subjects
Primary: 20C08: Hecke algebras and their representations
Secondary: 20G40: Linear algebraic groups over finite fields

Citation

Geck, Meinolf; Iancu, Lacrimioara. Lusztig's $a$-function in type $B_{n}$ in the asymptotic case. Nagoya Math. J. 182 (2006), 199--240. https://projecteuclid.org/euclid.nmj/1150810007


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References

  • S. Ariki, Robinson-Schensted correspondence and left cells , Combinatorial methods in representation theory (Kyoto, 1998), 1--20, Adv. Stud. Pure Math., 28, Kinokuniya, Tokyo (2000).
  • C. Bonnafé and L. Iancu, Left cells in type $B_n$ with unequal parameters , Represent. Theory, 7 (2003), 587--609.
  • C. Bonnafé, Two-sided cells in type $B$ in the asymptotic case , to appear, J. Algebra.
  • R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, New York (1985; reprinted 1993 as Wiley Classics Library Edition).
  • R. Dipper and G. D. James, Blocks and idempotents of Hecke algebras of general linear groups , Proc. London Math. Soc., 54 (1989), 57--82.
  • R. Dipper, G. D. James and G. E. Murphy, Hecke algebras of type $B_n$ at roots of unity , Proc. London Math. Soc., 70 (1995), 505--528.
  • R. Dipper, G. James and G. E. Murphy, Gram determinants of type $B_n$ , J. Algebra, 189 (1997), 481--505.
  • J. Du, B. Parshall, and L. Scott, Cells and $q$-Schur algebras , Transformation Groups, 3 (1998), 33--49.
  • W. Fulton, Young tableaux, London Math. Soc. Stud. Texts, vol. 35, Cambridge University Press (1997).
  • M. Geck, Constructible characters, leading coefficients and left cells for finite Coxeter groups with unequal parameters , Represent. Theory, 6 (2002), 1--30 (electronic).
  • M. Geck, Left cells and constructible representations , Represent. Theory, 9 (2005), 385--416.
  • M. Geck, Relative Kazhdan-Lusztig cells , (submitted); preprint available at arXiv.org/math.RT/0504216.
  • M. Geck, Kazhdan-Lusztig cells and the Murphy basis , to appear, Proc. London Math. Soc.
  • M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Math. Soc. Monographs, New Series 21, Oxford University Press (2000).
  • P. N. Hoefsmit, Representations of Hecke algebras of finite groups with BN pairs of classical type , Ph.D. thesis, University of British Columbia, Vancouver (1974).
  • D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math., 53 (1979), 165--184.
  • G. Lusztig, Left cells in Weyl groups , Lie Group Representations, I (R. L. R. Herb and J. Rosenberg, eds.), Lecture Notes in Math., vol. 1024, Springer-Verlag (1983), 99--111.
  • G. Lusztig, Cells in affine Weyl groups , Algebraic groups and related topics, Advanced Studies in Pure Math. 6, Kinokuniya and North-Holland (1985), 255--287.
  • G. Lusztig, Leading coefficients of character values of Hecke algebras , Proc. Symp. Pure Math. 47, Amer. Math. Soc., Providence, RI (1987), 235--262.
  • G. Lusztig, Hecke algebras with unequal parameters, CRM Monographs Ser. 18, Amer. Math. Soc., Providence, RI (2003).
  • G. Lusztig and N. Xi, Canonical left cells in affine Weyl groups , Advances in Math., 72 (1988), 284--288.
  • B. Srinivasan, A geometric approach to the Littlewood-Richardson rule , J. Algebra, 187 (1997), 227--235.