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), p. 707-720 / Harvested from Numdam

Résumé

Soient $G$ un groupe algébrique semi-simple complexe, ${\frak g}={\rm Lie}(G)$, $U$ l’algèbre enveloppante de ${\frak g}$, et $X$ la variété des drapeaux de $G$. Soit ${\frak h}$ une sous-algèbre de Cartan de ${\frak g}$. Pour $\mu \in {\frak h}^*$, soit $J_\mu $ l’idéal primitif minimal correspondant, soit $U_\mu =U/J_\mu $, et ${\cal T}_{U_\mu }:K_0(U_\mu )\rightarrow {\Bbb C}$ la trace de Hattori-Stallings. Des résultats de Hodges suggèrent d’étudier cette application en vue de classifier les ${\Bbb C}$-algèbres $U_\mu $ à isomorphisme ou équivalence de Morita près. Pour $\mu $ régulier, Hodges a montré que $K_0(U_\mu )\cong K_0(X)$. Dans ce cas, $K_0(U_\mu )$ est engendré par les classes correspondant aux fibrés en droites $G$-linéarisés sur $X$, et la valeur de ${\cal T}_{U_\mu }$ 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.

Abstract

Let $G$ be a semisimple complex algebraic group and $X$ its flag variety. Let ${\frak g}={\rm Lie}(G)$ and let $U$ be its enveloping algebra. Let ${\frak h}$ be a Cartan subalgebra of ${\frak g}$. For $\mu \in {\frak h}^*$, let $J_\mu $ be the corresponding minimal primitive ideal, let $U_\mu =U/J_\mu $, and let ${\cal T}_{U_\mu }:K_0(U_mu)\rightarrow {\Bbb C}$ 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 ${\Bbb C}$-algebras $U_\mu $. When $\mu $ is regular, Hodges has shown that $K_0(U_\mu )\cong K_0(X)$. In this case $K_0(U_\mu )$ is generated by the classes corresponding to $G$-linearized line bundles on $X$, and the value of ${\cal T}_{U_\mu }$ 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.

Détail

Publié le : 1995-01-01

MR 96i:17006 | Zbl 0818.17006

DOI : https://doi.org/10.5802/aif.1471

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