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Feb 21, 2022 at 14:00 history edited LSpice CC BY-SA 4.0
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May 5, 2017 at 1:41 history edited George McNinch CC BY-SA 3.0
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May 5, 2017 at 0:50 comment added George McNinch @Paul: Indeed, "zero locus of the positive degree invariants" is surely what I should have intended.
May 4, 2017 at 21:31 comment added Paul Levy I checked in Magma and the nullcone of $\mathfrak{sp}_4$ in characteristic 2 is reduced...
May 4, 2017 at 21:19 comment added Paul Levy By the scheme of nilpotent elements, I assume you mean the zero locus of the set of homogeneous invariants of positive degree. But in characteristic 2 I don't think the ring of invariants is known in general, which would suggest that your question is quite non-trivial. Incidentally, I am fairly sure one can show that the nullcone of $\mathfrak{pgl}_n$ is non-reduced at the subregular locus for arbitrary $n$.
May 4, 2017 at 17:05 comment added George McNinch Maybe a first question is: if $G$ is simply connected case, are the "scheme of unipotent elements" and the "scheme of nilpotent elements" reduced?
May 4, 2017 at 14:34 comment added Jim Humphreys This does get complicated, but I'm asking mainly about simply connected groups $G$ for bad $p$ (as in Springer's theorem for good $p$). Your discussion illustrates the extra obstacles when $G$ isn't simply connected. [By the way, I'd expect similar phenomena for adjoint groups of type $A_\ell$ whenever $p | (\ell+1)$, though of course it's hard to be as concrete as in the case $\ell =1$.]
May 4, 2017 at 12:29 history answered George McNinch CC BY-SA 3.0