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

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Patrick Polo
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, 45 no. 3 (1995), p. 707-720, doi: 10.5802/aif.1471
Article PDF | Reviews MR 96i:17006 | Zbl 0818.17006

Résumé - 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.

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