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

Rogawski’s conjecture on the Jantzen filtration for the degenerate affine Hecke algebra of type A
HTML articles powered by AMS MathViewer

by Takeshi Suzuki

Represent. Theory 2 (1998), 393-409

DOI: https://doi.org/10.1090/S1088-4165-98-00043-0

Published electronically: October 26, 1998

PDF | Request permission

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

T. Arakawa and T. Suzuki, Duality between $\mathfrak {sl}_n(\mathbb {C})$ and the degenerate Affine Hecke Algebra, to appear in Jour. of Alg.
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.
Dan Barbasch, Filtrations on Verma modules, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 489–494 (1984). MR 740080, DOI 10.24033/asens.1457
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
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, DOI 10.1090/advsov/016.1/01
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Differential operators on the base affine space and a study of ${\mathfrak {g}}$-modules, Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) Halsted, New York, 1975, pp. 21–64. MR 0578996
J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387–410. MR 632980, DOI 10.1007/BF01389272
Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
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, DOI 10.1007/BF01078042
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, DOI 10.1007/BF01077327
V. G. Drinfel′d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053, DOI 10.1007/BF01077318
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
O. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 261–302. MR 644519, DOI 10.24033/asens.1406
V. A. Ginzburg, A proof of the Deligne-Langlands conjecture, Dokl. Akad. Nauk SSSR 293 (1987), no. 2, 293–297 (Russian). MR 884035
Victor Ginzburg, Geometrical aspects of representation theory, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 840–848. MR 934285
Sergei Gelfand and Robert MacPherson, Verma modules and Schubert cells: a dictionary, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 34th Year (Paris, 1981) Lecture Notes in Math., vol. 924, Springer, Berlin-New York, 1982, pp. 1–50. MR 662251, DOI 10.1007/BFb0092924
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943, DOI 10.1007/BFb0069521
David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
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, DOI 10.1090/S1088-4165-02-00172-3
J. D. Rogawski, On modules over the Hecke algebra of a $p$-adic group, Invent. Math. 79 (1985), no. 3, 443–465. MR 782228, DOI 10.1007/BF01388516
A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. II. On irreducible representations of $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084, DOI 10.24033/asens.1379
A. Z. Zelevinsky, $p$-adic analogue of the Kazhdan-Lusztig Hypothesis, Funct. Anal. Appl. 15, No 2 (1981), 83-92.
A. Z. Zelevinsky, Two remarks on graded nilpotent classes, Russ. Math. Surveys 40, No 1 (1985), 249-250.
A. Z. Zelevinsky, Resolvents, dual pairs and character formulas, Functional Anal. Appl. 21 (1987), 152-154.