Multiplicity of the adjoint representation in simple quotients of the enveloping algebra of a simple Lie algebra
HTML articles powered by AMS MathViewer
- by Anthony Joseph PDF
- Trans. Amer. Math. Soc. 316 (1989), 447-491 Request permission
Abstract:
Let $\mathfrak {g}$ be a complex simple Lie algebra, $\mathfrak {h}$ a Cartan subalgebra and $U(\mathfrak {g})$ the enveloping algebra of $\mathfrak {g}$. We calculate for each maximal two-sided ideal ${J_{\max }}(\lambda ):\lambda \in {\mathfrak {h}^{\ast }}$ of $U(\mathfrak {g})$ the number of times the adjoint representation occurs in $U(\mathfrak {g})/{J_{\max }}(\lambda )$. This is achieved by reduction via the Kazhdan-Lusztig polynomials to the case when $\lambda$ lies on a corner, i.e. is a multiple of a fundamental weight. Remarkably in this case one can always present $U(\mathfrak {g})/{J_{\max }}(\lambda )$ as a (generalized) principal series module and here we also calculate its Goldie rank as a ring which is a question of independent interest. For some of the more intransigent cases it was necessary to use recent very precise results of Lusztig on left cells. The results are used to show how a recent theorem of Gupta established for "nonspecial" $\lambda$ can fail if $\lambda$ is singular. Finally we give a quite efficient procedure for testing if an induced ideal is maximal.References
- Dan Barbasch and David Vogan, Primitive ideals and orbital integrals in complex classical groups, Math. Ann. 259 (1982), no. 2, 153–199. MR 656661, DOI 10.1007/BF01457308
- Dan Barbasch and David Vogan, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 (1983), no. 2, 350–382. MR 691809, DOI 10.1016/0021-8693(83)90006-6
- W. M. Beynon and G. Lusztig, Some numerical results on the characters of exceptional Weyl groups, Math. Proc. Cambridge Philos. Soc. 84 (1978), no. 3, 417–426. MR 503002, DOI 10.1017/S0305004100055249
- Walter Borho and Jean-Luc Brylinski, Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules, Invent. Math. 69 (1982), no. 3, 437–476. MR 679767, DOI 10.1007/BF01389364
- Walter Borho and Jens Carsten Jantzen, Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-Algebra, Invent. Math. 39 (1977), no. 1, 1–53 (German, with English summary). MR 453826, DOI 10.1007/BF01695950
- Walter Borho and Robert MacPherson, Partial resolutions of nilpotent varieties, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 23–74. MR 737927
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
- Nicole Conze-Berline and Michel Duflo, Sur les représentations induites des groupes semi-simples complexes, Compositio Math. 34 (1977), no. 3, 307–336. MR 439991
- Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
- 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 R. K. Gupta, Copies of the adjoint representation inside induced ideals, preprint, Paris, 1985.
- Ryoshi Hotta, On Joseph’s construction of Weyl group representations, Tohoku Math. J. (2) 36 (1984), no. 1, 49–74. MR 733619, DOI 10.2748/tmj/1178228903
- Jens C. Jantzen, Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), no. 1, 53–65. MR 439902, DOI 10.1007/BF01391218
- Jens Carsten Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR 552943
- Jens Carsten Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 3, Springer-Verlag, Berlin, 1983 (German). MR 721170, DOI 10.1007/978-3-642-68955-0
- A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 1–29. MR 404366 —, Goldie rank in the enveloping algebra of a semi-simple Lie algebra. I, II, III, J. Algebra 65 (1980), 269-283; 284-306; 73 (1981), 295-326.
- A. Joseph, Kostant’s problem, Goldie rank and the Gel′fand-Kirillov conjecture, Invent. Math. 56 (1980), no. 3, 191–213. MR 561970, DOI 10.1007/BF01390044
- Anthony Joseph, On the variety of a highest weight module, J. Algebra 88 (1984), no. 1, 238–278. MR 741942, DOI 10.1016/0021-8693(84)90100-5
- Anthony Joseph, On the associated variety of a primitive ideal, J. Algebra 93 (1985), no. 2, 509–523. MR 786766, DOI 10.1016/0021-8693(85)90172-3
- Anthony Joseph, On the cyclicity of vectors associated with Duflo involutions, Noncommutative harmonic analysis and Lie groups (Marseille-Luminy, 1985) Lecture Notes in Math., vol. 1243, Springer, Berlin, 1987, pp. 144–188. MR 897541, DOI 10.1007/BFb0073021
- Anthony Joseph, A criterion for an ideal to be induced, J. Algebra 110 (1987), no. 2, 480–497. MR 910397, DOI 10.1016/0021-8693(87)90059-7
- A. Joseph, Dixmier’s problem for Verma and principal series submodules, J. London Math. Soc. (2) 20 (1979), no. 2, 193–204. MR 551445, DOI 10.1112/jlms/s2-20.2.193
- G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125–175. MR 463275, DOI 10.1007/BF01390002
- G. Lusztig, A class of irreducible representations of a Weyl group, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 3, 323–335. MR 546372
- George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
- George Lusztig, Sur les cellules gauches des groupes de Weyl, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 1, 5–8 (French, with English summary). MR 827096
- G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. (2) 19 (1979), no. 1, 41–52. MR 527733, DOI 10.1112/jlms/s2-19.1.41
- I. G. Macdonald, Some irreducible representations of Weyl groups, Bull. London Math. Soc. 4 (1972), 148–150. MR 320171, DOI 10.1112/blms/4.2.148
- Nicolas Spaltenstein, On the fixed point set of a unipotent element on the variety of Borel subgroups, Topology 16 (1977), no. 2, 203–204. MR 447423, DOI 10.1016/0040-9383(77)90022-2 T. Tanisaki, Private communication.
- A. Joseph, Completion functors in the ${\scr O}$ category, Noncommutative harmonic analysis and Lie groups (Marseille, 1982) Lecture Notes in Math., vol. 1020, Springer, Berlin, 1983, pp. 80–106. MR 733462, DOI 10.1007/BFb0071498
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 316 (1989), 447-491
- MSC: Primary 17B35; Secondary 22E47
- DOI: https://doi.org/10.1090/S0002-9947-1989-1020500-3
- MathSciNet review: 1020500