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, Volume 45 (1995) no. 3, p. 707-720

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 μ𝔥 * , 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) 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 -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.

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 μ𝔥 * , soit J μ l’idéal primitif minimal correspondant, soit U μ =U/J μ , et 𝒯 U μ :K 0 (U μ ) la trace de Hattori-Stallings. Des résultats de Hodges suggèrent d’étudier cette application en vue de classifier les -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.

@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {45},
     number = {3},
     year = {1995},
     pages = {707-720},
     doi = {10.5802/aif.1471},
     zbl = {0818.17006},
     mrnumber = {96i:17006},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1995__45_3_707_0}
}
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, Volume 45 (1995) no. 3, pp. 707-720. doi : 10.5802/aif.1471. http://www.numdam.org/item/AIF_1995__45_3_707_0/

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