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

Suzuki, Takeshi

doi:10.1090/S1088-4165-98-00043-0

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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
Posted: October 26, 1998
MathSciNet review: 1651408
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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.

[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] Dan Barbasch, Filtrations on Verma modules, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 489–494 (1984). MR 740080 (85j:17013)
[BB1] Alexandre Beĭlinson and Joseph Bernstein, Localisation de 𝑔-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15–18 (French, with English summary). MR 610137 (82k:14015)
[BB2] A. Beĭlinson and J. Bernstein, A proof of Jantzen conjectures, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 1–50. MR 1237825 (95a:22022)
[BGG] I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Differential operators on the base affine space and a study of 𝔤-modules, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 21–64. MR 0578996 (58 #28285)
[BK] J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980 (83e:22020), http://dx.doi.org/10.1007/BF01389272
[CG] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston Inc., Boston, MA, 1997. MR 1433132 (98i:22021)
[Ch1] I. V. Cherednik, An analogue of the character formula for Hecke algebras, Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 94–95 (Russian). MR 902309 (88m:22039)
[Ch2] I. V. Cherednik, Special bases of irreducible representations of a degenerate affine Hecke algebra, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 87–88 (Russian). MR 831062 (87m:22031)
[Dr] V. G. Drinfel′d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053 (87m:22044)
[GJ1] O. Gabber and A. Joseph, On the Bernšteĭn-Gel′fand-Gel′fand resolution and the Duflo sum formula, Compositio Math. 43 (1981), no. 1, 107–131. MR 631430 (82k:17009)
[GJ2] O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 261–302. MR 644519 (83e:17009)
[Gi1] V. A. Ginzburg, A proof of the Deligne-Langlands conjecture, Dokl. Akad. Nauk SSSR 293 (1987), no. 2, 293–297 (Russian). MR 884035 (89a:22031)
[Gi2] Victor Ginzburg, Geometrical aspects of representation theory, (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 840–848. MR 934285 (90e:22028)
[GM] Sergei Gelfand and Robert MacPherson, Verma modules and Schubert cells: a dictionary, (Paris, 1981) Lecture Notes in Math., vol. 924, Springer, Berlin, 1982, pp. 1–50. MR 662251 (84h:17004)
[Hu] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460 (92h:20002)
[Ja] Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943 (81m:17011)
[KL1] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412 (81j:20066), http://dx.doi.org/10.1007/BF01390031
[Lu1] George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016 (90e:16049), http://dx.doi.org/10.1090/S0894-0347-1989-0991016-9
[Lu2] George Lusztig, Cuspidal local systems and graded Hecke algebras. II, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 217–275. With errata for Part I [Inst. Hautes Études Sci. Publ. Math.\ No. 67 (1988), 145–202; MR0972345 (90e:22029)]. MR 1357201 (96m:22038), http://dx.doi.org/10.1090/S1088-4165-02-00172-3
[Ro] J. D. Rogawski, On modules over the Hecke algebra of a 𝑝-adic group, Invent. Math. 79 (1985), no. 3, 443–465. MR 782228 (86j:22028), http://dx.doi.org/10.1007/BF01388516
[Ze1] A. V. Zelevinsky, Induced representations of reductive 𝔭-adic groups. II. On irreducible representations of 𝔊𝔏(𝔫), Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084 (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.

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