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.

Soient $G$ un groupe algébrique semi-simple complexe, $𝔤=\mathrm{Lie}\left(G\right)$, $U$ l’algèbre enveloppante de $𝔤$, et $X$ la variété des drapeaux de $G$. Soit $𝔥$ une sous-algèbre de Cartan de $𝔤$. Pour $\mu \in {𝔥}^{*}$, soit $J_{\mu }$ l’idéal primitif minimal correspondant, soit $U_{\mu }=U/J_{\mu }$, et ${𝒯}_{U_{\mu }}:K_0(U_{\mu })\to ℂ$ la trace de Hattori-Stallings. Des résultats de Hodges suggèrent d’étudier cette application en vue de classifier les $ℂ$-algèbres $U_{\mu }$ à isomorphisme ou équivalence de Morita près. Pour $\mu$ régulier, Hodges a montré que $K_0(U_{\mu })\cong K_0\left(X\right)$. 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 ${𝒯}_{U_{\mu }}$ 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.

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

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