On the $K$-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case

Polo, Patrick

doi:10.5802/aif.1471

On the

K

-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case

Polo, Patrick

Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 707-720.

Résumé
Abstract

Soient $G$ un groupe algébrique semi-simple complexe, $𝔤 = Lie (G)$ , $U$ l’algèbre enveloppante de $𝔤$ , et $X$ la variété des drapeaux de $G$ . Soit $𝔥$ une sous-algèbre de Cartan de $𝔤$ . Pour $μ \in 𝔥^{*}$ , soit $J_{μ}$ l’idéal primitif minimal correspondant, soit $U_{μ} = U / J_{μ}$ , et $𝒯_{U_{μ}} : K_{0} (U_{μ}) \to ℂ$ la trace de Hattori-Stallings. Des résultats de Hodges suggèrent d’étudier cette application en vue de classifier les $ℂ$ -algèbres $U_{μ}$ à isomorphisme ou équivalence de Morita près. Pour $μ$ régulier, Hodges a montré que $K_{0} (U_{μ}) ≅ K_{0} (X)$ . Dans ce cas, $K_{0} (U_{μ})$ est engendré par les classes correspondant aux fibrés en droites $G$ -linéarisés sur $X$ , et la valeur de $𝒯_{U_{μ}}$ sur ces générateurs a été calculée par Hodges et Holland, dans un cas particulier, puis par Perets et l’auteur en général. Nous étendons ici ce résultat au cas singulier.

Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let $𝔤 = Lie (G)$ and let $U$ be its enveloping algebra. Let $𝔥$ be a Cartan subalgebra of $𝔤$ . For $μ \in 𝔥^{*}$ , let $J_{μ}$ be the corresponding minimal primitive ideal, let $U_{μ} = U / J_{μ}$ , and let $𝒯_{U_{μ}} : K_{0} (U_{m} u) \to ℂ$ be the Hattori-Stallings trace. Results of Hodges suggest to study this map as a step towards a classification, up to isomorphism or Morita equivalence, of the $ℂ$ -algebras $U_{μ}$ . When $μ$ is regular, Hodges has shown that $K_{0} (U_{μ}) ≅ K_{0} (X)$ . In this case $K_{0} (U_{μ})$ is generated by the classes corresponding to $G$ -linearized line bundles on $X$ , and the value of $𝒯_{U_{μ}}$ on these generators was computed by Hodges and Holland, in a special case, and then by Perets and the author, in general. This result is extended here to the singular case.

MR Zbl

DOI : 10.5802/aif.1471

@article{AIF_1995__45_3_707_0,
     author = {Polo, Patrick},
     title = {On the $K$-theory and {Hattori-Stallings} traces of minimal primitive factors of enveloping algebras of semisimple {Lie} algebras : the singular case},
     journal = {Annales de l'Institut Fourier},
     pages = {707--720},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     doi = {10.5802/aif.1471},
     zbl = {0818.17006},
     mrnumber = {96i:17006},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1471/}
}

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Polo, Patrick. On the $K$-theory and Hattori-Stallings traces of minimal primitive factors of enveloping algebras of semisimple Lie algebras : the singular case. Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 707-720. doi : 10.5802/aif.1471. https://aif.centre-mersenne.org/articles/10.5802/aif.1471/

Bibliographie
Cité par

[1] H. Bass, Algebraic K-theory, Benjamin, 1968. | MR | Zbl

[2] H. Bass, Euler Characteristics and Characters of Discrete Groups, Invent. Math., 35 (1976), 155-196. | MR | Zbl

[3] A. Beilinson, J. Bernstein, Localisation de g-modules, C. R. Acad. Sc. Paris, 292 (1981), 15-18. | MR | Zbl

[4] J. Bernstein, Trace in Categories, pp. 417-423 in : Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (A. Connes et al., eds.), Birkhäuser, 1990. | MR | Zbl

[5] J. Bernstein, S.I. Gelfand, Tensor products of finite and infinite dimensional representations of semisimple Lie algebras, Compositio Math., 41 (1980), 245-285. | Numdam | MR | Zbl

[6] A. Borel et al., Algebraic D-modules, Academic Press, 1987. | MR | Zbl

[7] N. Bourbaki, Groupes et algèbres de Lie, Chap. IV-VI, Hermann, 1968.

[8] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math., 21 (1973), 287-301. | MR | Zbl

[9] J. Dixmier, Algèbres Enveloppantes, Gauthier-Villars, 1974. | MR | Zbl

[10] H. Hecht, D. Miličić, W. Schmid, J.A. Wolf, Localization and standard modules for real semisimple Lie groups I : The duality theorem, Invent. Math., 90 (1987), 297-332. | MR | Zbl

[11] T.J. Hodges, K-Theory of D-modules and primitive factors of enveloping algebras of semisimple Lie algebras, Bull. Sc. Math., 113 (1989), 85-88. | MR | Zbl

[12] T.J. Hodges, Morita Equivalence of Primitive Factors of U(sl(2)), pp. 175-179 in : Kazhdan-Lusztig Theory and Related Topics (V. Deodhar, ed.), Contemporary Math. 139 (1992). | MR | Zbl

[13] T.J. Hodges, M.P. Holland, Chern characters, reduced ranks and D-modules on the flag variety, Proc. Edinburgh Math. Soc., 37 (1994), 477-482. | MR | Zbl

[14] T.J. Hodges, S.P. Smith, On the global dimension of certain primitive factors of the enveloping algebra of a semi-simple Lie algebra, J. London Math. Soc., 32 (1985), 411-418. | MR | Zbl

[15] M.P. Holland, P. Polo, K-theory of twisted differential operators on flag varieties, preprint (Dec. 1994). | Zbl

[16] J.C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math. 750, Springer-Verlag, 1979. | MR | Zbl

[17] J.C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Springer-Verlag, 1983. | Zbl

[18] A. Joseph, J.T. Stafford, Modules of t-finite vectors over semi-simple Lie algebras, Proc. London Math. Soc., 49 (1984), 361-384. | MR | Zbl

[19] M. Kashiwara, Representation theory and D-modules on flag varieties, pp. 55-109 in : Orbites unipotentes et représentations III (éd. M. Andler), Astérisque, 173-174 (1989). | MR | Zbl

[20] R. Marlin, Anneaux de Grothendieck des variétés de drapeaux, Bull. Soc. Math. France, 104 (1976), 337-348. | Numdam | MR | Zbl

[21] G.S. Perets, P. Polo, On the Hattori-Stallings trace for certain primitive factors of enveloping algebras of semisimple Lie algebras, Math. Z. (to appear). | Zbl

[22] D. Quillen, Higher algebraic K-theory, pp. 85-147 in : Algebraic K-Theory I (H. Bass, ed.), Lecture Notes in Math. 341, Springer-Verlag, 1973. | MR | Zbl

[23] W. Soergel, Universelle versus relative Einhüllende : Eine geometrische Untersuchung von Quotienten von universellen Einhüllende halbeinfacher Lie-Algebren, Math. Annalen, 284 (1989), 177-198. | MR | Zbl

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