Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165



Rogawski's conjecture on the Jantzen
filtration for the degenerate affine
Hecke algebra of type $A$

Author: Takeshi Suzuki
Journal: Represent. Theory 2 (1998), 393-409
MSC (1991): Primary 22E50; Secondary 17B10
Published electronically: October 26, 1998
MathSciNet review: 1651408
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The functors constructed by Arakawa and the author relate the representation theory of ${\mathfrak{gl}}_n$ and that of the degenerate affine Hecke algebra $H_{\ell}$ of $\mathrm{GL}_{\ell}$. They transform the Verma modules over ${\mathfrak{gl}}_n$ to the standard modules over $H_{\ell}$. In this paper we prove that they transform the simple modules to the simple modules (in more general situations than in the previous paper). We also prove that they transform the Jantzen filtration on the Verma modules to that on the standard modules. We obtain the following results for the representations of $H_{\ell}$ by translating the corresponding results for ${\mathfrak{gl}}_n$ through the functors: (i) the (generalized) Bernstein-Gelfand-Gelfand resolution for a certain class of simple modules, (ii) the multiplicity formula for the composition series of the standard modules, and (iii) its refinement concerning the Jantzen filtration on the standard modules, which was conjectured by Rogawski.

References [Enhancements On Off] (What's this?)

  • [AS] T. Arakawa and T. Suzuki, Duality between $\mathfrak{sl}_n({\mathbb C})$ and the degenerate Affine Hecke Algebra, to appear in Jour. of Alg.
  • [AST] T. Arakawa, T. Suzuki and A. Tsuchiya, Degenerate double affine Hecke algebras and conformal field theory, to appear in Topological Field Theory, Primitive Forms and Related Topics; the Proceedings of the 38$^{th}$ Taniguchi Symposium, Birkhäuser.
  • [Ba] D. Barbasch, Filtrations on Verma modules, Ann. Sci. Ecole Norm. Sup., $4^e$ Serie 16 (1984), 489-494. MR 85j:17013
  • [BB1] A. Beilinson and J. Bernstein [I. N. Bernstein], Localisation de $\mathfrak{g}$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 15-18. MR 82k:14015
  • [BB2] A. Beilinson and I. N. Bernstein, A proof of Jantzen conjecture, Adv. in Soviet Math. 16, Part 1 (1993), 1-50. MR 95a:22022
  • [BGG] I. N. Bernstein, I. M. Gel'fand and S. I. Gel'fand, Differential operators on the base affine space and a study of $\mathfrak{g}$ modules, In Lie groups and their representations (Proc. Summer School, Boyai Janos Math. Soc., Budapest, 1971, I. M. Gelfand, Ed.), London, Hilger, (1975). MR 58:28285
  • [BK] J. L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), 387-410. MR 83e:22020
  • [CG] N. Chriss and V. Ginzburg, Representation theory and complex geometry. Birkhäuser, (1997). MR 98i:22021
  • [Ch1] I. V. Cherednik, An analogue of the character formulas for Hecke algebras, Funct. Anal. Appl. 21, No 2 (1987), 94-95. MR 88m:22039
  • [Ch2] I. V. Cherednik, Special bases of irreducible representations of a degenerate affine Hecke algebra, Funct. Anal. Appl. 20, No 1 (1986), 87-88. MR 87m:22031
  • [Dr] V. G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20, No 1 (1986), 67-70. MR 87m:22044
  • [GJ1] O. Gabber and A. Joseph, On the Bernstein-Gelfand-Gelfand resolution and the Duflo sum formula, Compositio Math. 43, (1981), 107-131. MR 82k:17009
  • [GJ2] O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. Ecole. Norm. Sup. (4) 14, (1981), 261-302. MR 83e:17009
  • [Gi1] V. A. Ginzburg, Proof of the Deligne-Langlands conjecture, Soviet. Math. Dokl. 293, No 2 (1987), 293-297. MR 89a:22031
  • [Gi2] V. A. Ginzburg, Geometric aspects of representation theory in Proceedings of ICM 1986, Berkeley, (1986), 840-848. MR 90e:22028
  • [GM] S. Gelfand and R. MacPherson, Verma modules and Schubert cells: a dictionary, Lecture Notes in Math., vol 924, (1982), Springer, 1-50. MR 84h:17004
  • [Hu] J. E. Humphreys, Reflection groups and Coxeter groups, (1990), Cambridge University Press. MR 92h:20002
  • [Ja] J. C. Jantzen, Moduln mit einem hochsten Gewicht. Lecture Note in Mathematics, vol 750, (1980), Springer, Berlin. MR 81m:17011
  • [KL1] D. Kazhdan and G. Lusztig, Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165-184. MR 81j:20066
  • [Lu1] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2, No 3 (1989), 599-635. MR 90e:16049
  • [Lu2] G. Lusztig, Cuspidal local systems and graded Hecke algebras, II, Representations of Groups; CMS Conf. Proc. 16, (1995), Amer. Math. Soc., 217-275. MR 96m:22038
  • [Ro] J. D. Rogawski, On modules over the Hecke algebra of a $p$-adic group, Invent. Math. 79 (1985), 443-465. MR 86j:22028
  • [Ze1] A. Z. Zelevinsky, Induced representations of reductive $p$-adic groups II, Ann. Sci. Ecole Norm. Sup. (4) Serie 13 (1980), 165-210. MR 83g:22012
  • [Ze2] A. Z. Zelevinsky, $p$-adic analogue of the Kazhdan-Lusztig Hypothesis, Funct. Anal. Appl. 15, No 2 (1981), 83-92.
  • [Ze3] A. Z. Zelevinsky, Two remarks on graded nilpotent classes, Russ. Math. Surveys 40, No 1 (1985), 249-250.
  • [Ze4] A. Z. Zelevinsky, Resolvents, dual pairs and character formulas, Functional Anal. Appl. 21 (1987), 152-154.

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (1991): 22E50, 17B10

Retrieve articles in all journals with MSC (1991): 22E50, 17B10

Additional Information

Takeshi Suzuki
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Japan

Received by editor(s): January 23, 1998
Received by editor(s) in revised form: August 31, 1998
Published electronically: October 26, 1998
Additional Notes: The author is supported by the JSPS Research Fellowships for Young Scientists.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society