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Structure of Modules Induced from Simple Modules with Minimal Annihilator

Published online by Cambridge University Press:  20 November 2018

Oleksandr Khomenko  and
Volodymyr Mazorchuk
Oleksandr Khomenko
Affiliation:
Mathematisches Institut der Universität Freiburg, Eckerstrasse 1, D-79104, Freiburg im Breisgau, FRG email: [email protected]
Volodymyr Mazorchuk
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE 751 06, Uppsala, Sweden email: [email protected]
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Abstract

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We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

Keywords

17B1022E47parabolic inductiongeneralized Verma modulesimple moduleHarish-Chandra bimoduleequivalent categories
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 2 , 01 April 2004 , pp. 293 - 309
Copyright
Copyright © Canadian Mathematical Society 2004

References

[Ba] Backelin, E., Representation of the category in Whittaker categories. Internat. Math. Res. Notices 1997, 153–172.Google Scholar
[BB] Beilinson, A., Bernstein, J., Localisation de g-modules. (French) C. R. Acad. Sci. Paris Ser. I Math. 292 (1981), no. 1, 15–18.Google Scholar
[BeGi] Beilinson, A., Ginzburg, V., Wall-crossing functors and D-modules. Represent. Theory 3 (1999), 1–31 (electronic).Google Scholar
[BG] Bernstein, J. N., Gelfand, S. I., Tensor products of finite- and Infinite-dimensional representations of semisimple Lie algebras. Compositio Math. 41 (1980), 245–285.Google Scholar
[BGG] Bernstein, I. N., Gelfand, I. M., Gelfand, S. I., The structure of representations generated by vectors of highest weight, Funkt. Anal. i Prilozhen. 5 (1971), 1–9.Google Scholar
[BK] Brylinski, J.-L.; Kashiwara, M., Kazhdan-Lusztig conjecture and holonomic systems. Invent. Math. 64 (1981), no. 3, 387–410.Google Scholar
[CPS] Cline, E., Parshall, B., Scott, L., Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391 (1988), 85–99.Google Scholar
[CF] Coleman, A. J. and Futorny, V. M., Stratièd L-modules, J. Algebra, 163 (1994), 219–234.Google Scholar
[D] Dixmier, J., Algebres enveloppantes, Gauthier-Villars, Paris, 1974.Google Scholar
[Dl] Dlab, V., Properly stratified algebras. C. R. Acad. Sci. Paris Ser. I Math. 331 (2000), no. 3, 191–196.Google Scholar
[DFO] Drozd, Yu. A., Futorny, V. M., Ovsienko, S. A., S-homomorphism of Harish-Chandra and G-modules generated by semiprimitive elements, Ukrainian Math. J. 42 (1990), 1032–1037.Google Scholar
[FKM1] Futorny, V., König, S. and Mazorchuk, V., Categories of induced modules and standardly stratièd algebras, Alg. and Repr. Theory 5 (2002), 259–276.Google Scholar
[FKM2] Futorny, V., König, S. and Mazorchuk, V., S-subcategories in . Manuscripta Math. 102 (2000), no. 4, 487–503.Google Scholar
[FM] Futorny, V., Mazorchuk, V., Structure of α-Stratièd Modules for Finite-Dimensional Lie Algebras, Journal of Algebra, I, 183 (1996), 456–482.Google Scholar
[GJ] Gabber, O., Joseph, A., Towards the Kazhdan-Lusztig conjecture. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 261–302.Google Scholar
[GM] Gomez, X., Mazorchuk, V., On an analogue of BGG-reciprocity. Comm. Algebra 29 (2001), no. 12, 5329–5334.Google Scholar
[J1] Jantzen, J. C., Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren. Math. Ann. 226 (1977), no. 1, 53–65.Google Scholar
[J2] Jantzen, J. C., Einhüllende Algebren halbeinfacher Lie-Algebren. (German) [Enveloping algebras of semisimple Lie algebras] Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 3. Springer-Verlag, Berlin-New York, 1983.Google Scholar
[KM1] Khomenko, A., Mazorchuk, V., On the determinant of Shapovalov form for Generalized Verma modules, J. Algebra 215 (1999), 318–329.Google Scholar
[KM2] Khomenko, A., Mazorchuk, V., A note on simplicity of Generalized Verma modules, Commentarii Mathematici Universitatis Sancti Pauli 48 (1999), 145–148.Google Scholar
[KM3] Khomenko, A., Mazorchuk, V., Generalized Verma modules induced from sl(2; ℂ) and associated Verma modules. J. Algebra 242 (2001), no. 2, 561–576.Google Scholar
[KM4] Khomenko, A., Mazorchuk, V., Rigidity of generalized Verma modules. Coloq. Math 92 (2002), no. 1, 45–57.Google Scholar
[KM5] Khomenko, A., Mazorchuk, V., On multiplicities of simple subquotients in generalized Verma modules, Czechoslovak Math. J., 53 (2002), 337–343.Google Scholar
[KoM1] König, S. and Mazorchuk, V., Enright's completions and injectively copresented modules, Trans. Amer. Math. Soc. 354 (2002), 2725–2743.Google Scholar
[KoM2] König, S. and Mazorchuk, V., An equivalence of two categories of sl(n; C)-modules, Alg. and Repr. Theory 5 (2002), 319–329.Google Scholar
[MO] Mazorchuk, V. and Ovsienko, S., Submodule structure of Generalized Verma Modules induced from generic Gelfand-Zetlin modules, Alg. and Repr. Theory 1 (1998), 3–26.Google Scholar
[Mc1] McDowell, E., A module induced from a Whittaker module. Proc. Amer. Math. Soc. 118 (1993), 349–354.Google Scholar
[Mc2] McDowell, E., On modules induced from Whittaker modules. J. Algebra 96 (1985), 161–177.Google Scholar
[MS1] Miličić, D., Soergel, W., The composition series of modules induced from Whittaker modules. Comment. Math. Helv. 72 (1997), 503–520.Google Scholar
[MS2] Miličić, D., Soergel, W., Twisted Harish-Chandra sheaves and Whittaker modules; Preprint, Freiburg University, 1995.Google Scholar
[R] Rocha-Caridi, A., Splitting criteria for g-modules induced from a parabolic and a Bernstein – Gelfand – Gelfand resolution of a finite-dimensional, irreducible g-module. Trans. Amer. Math. Soc., 262 (1980), 335–366.Google Scholar
[Se] Semikhatov, A., Past the highest-weight, and what you can find there. New developments in quantum field theory (Zakopane, 1997), 329–339,Google Scholar
[S1] Soergel, W., Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe. (German) [Category O, perverse sheaves and modules over the coinvariants for the Weyl group] J. Amer.Math. Soc. 3 (1990), no. 2, 421–445.Google Scholar
[St] Stafford, J. T., Nonholonomic modules over Weyl algebras and enveloping algebras. Invent. Math. 79 (1985), no. 3, 619–638.Google Scholar