Skip to main content Accesibility Help
×
×
Home

KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS

  • JOHAN KÅHRSTRÖM (a1)
Abstract

Let be a finite dimensional complex semi-simple Lie algebra with Weyl group W and simple reflections S. For IS let I be the corresponding semi-simple subalgebra of . Denote by WI the Weyl group of I and let w and wI be the longest elements of W and WI, respectively. In this paper we show that the answer to Kostant's problem, i.e. whether the universal enveloping algebra surjects onto the space of all ad-finite linear transformations of a given module, is the same for the simple highest weight I-module LI(x) of highest weight x ⋅ 0, xWI, as the answer for the simple highest weight -module L(xwIw) of highest weight xwIw ⋅ 0. We also give a new description of the unique quasi-simple quotient of the Verma module Δ(e) with the same annihilator as L(y), yW.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      KOSTANT'S PROBLEM AND PARABOLIC SUBGROUPS
      Available formats
      ×
Copyright
References
Hide All
1.Andersen, H. H. and Stroppel, C., Twisting functors on , Represent. Theory 7 (2003), 681699.
2.Bernšteĭn, J. N. and Gelfand, S. I., Tensor products of finite- and infinite-dimensional representations of semisimple Lie algebras, Compositio Math. 41 (2) (1980), 245285.
3.Bernšteĭn, I. N., Gelfand, I. M. and Gelfand, S. I., A certain category of -modules. Funkcional. Anal. i Prilozen. 10 (2) (1976), 18.
4.Conze, N., Algèbres d'opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France 102 (1974), 379415.
5.Conze-Berline, N. and Duflo, M., Sur les représentations induites des groupes semi-simples complexes. Compositio Math. 34 (3) (1977), 307336.
6.Dixmier, J., Enveloping algebras, Graduate Studies in Mathematics, 11, (American Mathematical Society, Prodivence, RI, 1996).
7.Duflo, M., Sur la classification des idéaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semi-simple, Ann. Math. 105 (1977), 107120.
8.Gabber, O. and Joseph, A., On the Bernstein–Gelfand–Gelfand resolution and the Duflo Sum formula, Compositio Math. 43 (Fasc. 1) 1981, 108131.
9.Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Notes Series, Vol. 119 (Cambridge University Press, Cambridge, UK, 1988).
10.Humphreys, J., Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics 94 (American Mathematical Society, Providence, RI, 2008).
11.Jantzen, J. C., Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, Vol. 750 (Springer-Verlag, Berlin, 1979), ii195.
12.Jantzen, J. C., Einhüllende Algebren halbeinfacher Lie-Algebren, u Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 3 (Springer-Verlag, Berlin, 1983).
13.Joseph, A., A characteristic variety for the primitive spectrum of a semisimple Lie algebra, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1976), 102118. Lecture Notes in Mathematics, Vol. 587, Springer, Berlin, 1977.
14.Joseph, A., Dixmier's problem for Verma and principal series submodules, J. Lond. Math. Soc. 20 (1979), 193204.
15.Joseph, A., Kostant's problem, Goldie rank and the Gelfand–Kirillov conjecture, Invent. Math. 56 (1980), 191213.
16.Joseph, A., Goldie rank in the enveloping algebra of a semisimple Lie algebra, I, J. Algebra 65 (1980), 269283.
17.Joseph, A., Kostant's problem and Goldie rank, Noncommutative harmonic analysis and Lie groups (Marseille, 1980), 249266, Lecture Notes in Mathematics, 880 (Springer, Berlin-New York, 1981).
18.Joseph, A., A sum rule for scale factors in the Goldie rank polynomials, J. Algebra 118 (2) (1988), 276311.
19.Kåhrström, J., Mazorchuk, V., A new approach to Kostant's problem, preprint arXiv:0712.3117.
20.Kazhdan, D. and Lusztig, G., Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (2) (1979), 165184.
21.Khomenko, O. and Mazorchuk, V., Structure of modules induced from simple modules with minimal annihilator, Can. J. Math. 56 (2) (2004), 293309.
22.Khomenko, O. and Mazorchuk, V., On Arkhipov's and Enright's functors, Math. Z. 249 (2) (2005), 357386.
23.Lusztig, G., Characters of reductive groups over a finite field, in Annals of Mathematics Studies, Vol. 107 (Princeton University Press, Princeton, NJ, 1984).
24.Lusztig, G., Cells in affine Weyl groups, in Advanced Studies in Pure Mathematics, Vol. 6 (North-Holland, Amsterdam, 1985).
25.Lusztig, G., Cells in affine Weyl groups, II, Algebra 109 (2) (1987), 536548.
26.Mazorchuk, V., A twisted approach to Kostant's problem, Glasgow Math. J. 47 (2005), 549561.
27.Mazorchuk, V. and Stroppel, C., Categorification of (induced) cell modules and the rough structure of generalised Verma modules, Adv. Math. 219 (4) (2008), 13631426.
28.Mazorchuk, V. and Stroppel, C., Categorification of Wedderburn's basis for [Sn], Arch. Math. (Basel) 91 (1) (2008), 111.
29.Miličić, D. and Soergel, W., The composition series of modules induced from Whittaker modules, Comment. Math. Helv. 72 (4) (1997), 503520.
30.Stroppel, C., Category : Gradings and translation functors, J. Algebra 268 (1) (2003), 301326.
31.Stroppel, C., Composition factors of quotients of the universal enveloping algebra by primitive ideals, J. Lond. Math. Soc. (2) 70 (3) (2004), 643658.
32.Vogan, D., Ordering of the primitive spectrum of a semisimple Lie algebra, Math. Ann. 248 (3) (1980), 195203.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed