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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...