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Multiplicity-free primitive ideals associated with rigid nilpotent orbits. (English) Zbl 1353.17009

Let \(G\) be a complex simple algebraic group and denote by \(\mathfrak{g}\) the Lie algebra of \(G\). For a nilpotent element \(e\) of \(\mathfrak{g}\) denote by \(U(\mathfrak{g},e)\) the finite \(W\)-algebra associated with \((\mathfrak{g},e)\) and by \(\mathcal{E}\) the set of all one-dimensional representations of \(U(\mathfrak{g},e)\). Moreover, let \(G_e\) be the centralizer of \(e\) in \(G\). In the main result of the paper under review the author proves that the subset \(\mathcal{E}^\Gamma\) of \(\mathcal{E}\) consisting of all elements fixed by the action of the component group \(\Gamma\) of \(G_e\) on \(\mathcal{E}\) is non-empty. In particular, all finite \(W\)-algebras associated with \(\mathfrak{g}\) admit one-dimensional representations. For rigid nilpotent elements in exceptional Lie algebras he find irreducible highest weight \(\mathfrak{g}\)-module whose annihilators in the universal enveloping algebra \(U(\mathfrak{g})\) of \(\mathfrak{g}\) come from one-dimensional representations of \(U(\mathfrak{g},e)\) via Skryabin’s equivalence. It is then proved that the Zariski closure of every nilpotent orbit in \(\mathfrak{g}\) is the associated variety of a multiplicity-free (and so completely prime) primitive ideal of \(U(\mathfrak{g})\).

MSC:

17B35 Universal enveloping (super)algebras
17B08 Coadjoint orbits; nilpotent varieties
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)

Software:

PyCox
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References:

[1] Barbasch, W; Vogan, D, Primitive ideals and orbital integrals in classical groups, Math. Ann., 259, 153-199, (1982) · Zbl 0489.22010
[2] W. Barbasch, D. Vogan, Unipotent representations of complex semisimple groups, Ann. of Math. (2) 121 (1985), no. 1, 41-110. · Zbl 0582.22007
[3] Borho, W; Kraft, H, Über die Gelfand-Kirillov-dimension, Math. Ann., 220, 1-24, (1976) · Zbl 0306.17005
[4] N. Bourbaki, Groups et Algèbres de Lie, Chapitres IV, V, VI, Hermann, Paris, 1968. · Zbl 1005.17007
[5] Brown, J; Goodwin, S, Finite dimensional irreducible representations of finite W-algebras associated to even multiplicity nilpotent orbits in classical Lie algebras, Math. Z., 273, 123-160, (2013) · Zbl 1266.17003
[6] J. Brundan, S. Goodwin, A. Kleshchev, Highest weight theory for finite W-algebras, IMRN no. 15 (2008), Art. ID rnn051, 53 pp. · Zbl 1211.17024
[7] R. Brylinski, Dixmier algebras for classical complex nilpotent orbits via Kraft-Procesi models, in: The Orbit Method in Geometry and Physics (Marseille, 2000), Progress in Mathematics, Vol. 213, Birkhäuser, Boston, MA, 2003, pp. 49-67. · Zbl 1069.17005
[8] R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Pure and Applied Mathematics, Wiley, New York, 1985. · Zbl 0567.20023
[9] Chen, Y, Left cells in the Weyl group of type E_{8}, J. Algebra, 230, 805-830, (2000) · Zbl 0968.20022
[10] Chen, Y; Shi, J-Y, Left cells in the Weyl group of type E_{7}, Comm. Algebra, 26, 3837-3852, (1998) · Zbl 0916.20027
[11] D. H. Collingwood, W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993. · Zbl 0972.17008
[12] W. A. de Graaf, Computations with nilpotent elements in SLA, arXiv:1301.1149v1.
[13] J. Dixmier, Algèbres Enveloppantes, Gauthier-Villars, Paris, Bruxelles, Montréal, 1974. Russian transl.:Ж. Диксмье, Унuвeрсальныe обeрmывающue aлгeбры, Mir, Mир, M., 1978. · Zbl 0308.17007
[14] Duflo, M, Sur la classification des idéaux primitifs dans l’algèbre enveloppante d’une algèbre de Lie semi-simple, Ann. of Math., 105, 107-130, (1977) · Zbl 0346.17011
[15] Duflo, M, Représentations unitaires irréducibles des groupes semi-simples complexes de rang deux, Bull. Soc. Math. France, 107, 55-96, (1979) · Zbl 0407.22014
[16] Gan, WL; Ginzburg, V, Quantization of slodowy slices, Int. Math. Res. Not., 5, 243-255, (2002) · Zbl 0989.17014
[17] Geck, M, Pycox: computing with (finite) Coxeter groups and Iwahori-Hecke algebras, LMS J. Comput. Math., 15, 231-256, (2012) · Zbl 1296.20009
[18] Ginzburg, V, Harish-chandra bimodules for quantized slodowy slices, Represent. Theory, 13, 236-371, (2009) · Zbl 1250.17007
[19] Goodwin, S; Röhrle, G; Ubly, G, On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type, LMS J. Comput. Math., 13, 357-369, (2010) · Zbl 1278.17015
[20] Humphreys, JE, Modular representations of simple Lie algebras, Bull. Amer. Math. Soc. (N.S.), 35, 105-122, (1998) · Zbl 0962.17013
[21] J. E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category\( \mathcal{O} \), Grad. Stud. Math., Vol. 94, Amer. Math. Soc., Providence, RI, 2008. · Zbl 1177.17001
[22] J. C. Jantzen, Einhülende Algebren Halbeinfacher Lie-Algebren, Ergebnisse der Math., Vol. 3, Springer, New York, 1983. · Zbl 0541.17001
[23] J. C. Jantzen, Nilpotent orbits in representation theory, in: Lie Theory. Lie Algebras and Representations, Progress in Mathematics, Vol. 228, Birkhäuser, Boston, Boston, MA, 2004, pp. 1-211. · Zbl 1169.14319
[24] A. Joseph, Gelfand-Kirillov dimension for the annihilators of simple quotients of Verma modules, J. London Math. Soc. (2) 18 no. 1 (1978), 50-60. · Zbl 0401.17007
[25] A. Joseph, Primitive ideals in enveloping algebras, in: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 403-414. · Zbl 0916.20027
[26] A. Joseph, Kostant’s problem and Goldie rank, in: Non-commutative Harmonic Analysis and Lie Groups, Lecture Notes in Mathematics, Vol. 880, Springer-Verlag, Berlin, 1981, pp. 249-266. · Zbl 0489.22010
[27] Kazhdan, D; Lusztig, G, Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, 165-184, (1979) · Zbl 0499.20035
[28] R. Lawther, D. M. Testerman, Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups, Mem. Amer. Math. Soc. 210 (2011), no. 980. · Zbl 1266.20061
[29] Levasseur, T; Smith, SP, Primitive ideals and nilpotent orbits in type G_{2}, J. Algebra, 114, 81-105, (1988) · Zbl 0644.17005
[30] Losev, IV, Quantized symplectic actions and W-algebras, J. Amer. Math. Soc., 23, 35-59, (2010) · Zbl 1246.17015
[31] Losev, IV, Finite dimensional representations of W-algebras, Duke Math. J., 159, 99-143, (2011) · Zbl 1235.17007
[32] Losev, IV, 1-dimensional representations and parabolic induction for W-algebras, Adv. Math., 226, 4841-4883, (2011) · Zbl 1287.17025
[33] Losev, IV, On the structure of the category \( \mathcal{O} \) for W-algebras, Séminaires et Congrès, 24, 351-368, (2012)
[34] W. M. McGovern, Completely Prime Maximal Ideals and Quantization, Mem. Amer. Math. Soc. 108 (1994) no. 519. · Zbl 0820.17010
[35] Mœglin, C, Idéaux complètement premiers de l’algèbre enveloppante de \( {\mathfrak g} {\mathfrak l} \)_{n}(ℂ), J. Algebra, 106, 287-366, (1987) · Zbl 0637.17005
[36] Premet, A, Special transverse slices and their enveloping algebras, Adv. Math., 170, 1-55, (2002) · Zbl 1005.17007
[37] Premet, A, Enveloping algebras of slodowy slices and the Joseph ideal, J. Eur. Math. Soc., 9, 487-543, (2007) · Zbl 1134.17307
[38] Premet, A, Primitive ideals, non-restricted representations and finite W-algebras, Mosc. Math. J., 7, 743-762, (2007) · Zbl 1139.17005
[39] Premet, A, Commutative quotients of finite W-algebras, Adv. Math., 225, 269-306, (2010) · Zbl 1241.17015
[40] Premet, A, Enveloping algebras of slodowy slices and Goldie rank, Transform. Groups, 16, 857-888, (2011) · Zbl 1261.17011
[41] A. Premet, One-dimensional representations of finite W-algebras and Humphreys’ conjecture, in preparation.
[42] A. Premet, L. Topley, Derived subalgebras of centralisers and finite W-algebras, Compositio Math., to appear, arXiv:1301.4653v2. · Zbl 1345.17010
[43] S. Skryabin, A category equivalence, Appendix to [36]. · Zbl 1134.17307
[44] G. Ubly, A Computational Approach to 1-dimensional Representations of Finite W-algebras Associated with Simple Lie Algebras of Exceptional Type, PhD thesis, University of Southampton, ePrints, Soton, 2010, 177 pp., http://eprints.soton.ac.uk. · Zbl 1278.17015
[45] D. Vogan, The orbit method and primitive ideals for semisimple Lie algebras, in: Lie Algebras and Related Topics (Windsor, Ont. 1984), CMS Conf. Proc. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 281-316. · Zbl 0968.20022
[46] Yakimova, O, On the derived algebra of a centraliser, Bull. Sci. Math., 134, 579-587, (2010) · Zbl 1225.17012
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