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Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ not isomorphic to $sl(n)$. Let $\mathcal O$ be the minimal nilpotent orbit in $\mathfrak g^*$. Joseph proved that there exists unique two-sided ideal $J$ in $U(\mathfrak g)$ whose associated graded consists of functions which vanish on $\mathcal O$.

${\bf Question:}$ Is it known for which $\mathfrak g$ the algebra $U(\mathfrak g)/J$ has finite homological dimension?

It seems to me that it is so for $sp(4)$ but not so for $so(8)$ (but I didn't check the details).

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    $\begingroup$ I'm not sure about the answer, but maybe it's included in Jantzen's 1983 Ergebnisse volume on primitive ideals. Anyway, I guess you refer to the 1976 paper by Joseph numdam.org/numdam-bin/fitem?id=ASENS_1976_4_9_1_1_0 $\endgroup$
    – Jim Humphreys
    Commented Jul 13, 2015 at 23:42
  • $\begingroup$ I think Tony doesn't know the answer (and this question is definitely not discussed in his 1976 paper). At first I thought that the homological dimension should always be finite but it now seems to me that for $so(8)$ I have a very roundabout argument which proves that it can't be so... $\endgroup$
    – Alexander Braverman
    Commented Jul 14, 2015 at 5:34
  • $\begingroup$ @Sasha: I mainly wanted to confirm that the 1976 paper was your starting point. I agree that Joseph and Jantzen don't deal directly with finiteness of homological dimension; their main concern always seemed to be Gelfand-Kirillov dimension in the primitive ideal setting. $\endgroup$
    – Jim Humphreys
    Commented Jul 14, 2015 at 13:39

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