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Apr 17, 2013 at 13:25 comment added Jim Humphreys Concerning the field, the default assumption is that the characteristic not be 2. I'm not sure what's been done by now, but a simple search of MathSciNet shows at least 20 relevant items in the subject area 20G including Springer's last papers and a mini-workshop at Oberwolfach in 2008. By now there are quite a few directions of research, involving algebraic geometry and partly inspired by classical symmetric spaces of Lie groups, so it's worth searching the literature. (However, the type of examples you want may not be there yet.)
Apr 17, 2013 at 2:46 history edited Michael Joyce CC BY-SA 3.0
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Apr 17, 2013 at 2:43 comment added Michael Joyce I am generally considering an algebraically closed field, and I am fine with characteristic zero assumptions if necessary. I don't think reductive adds anything important, but I tried to cast as wide a net as possible while fishing for examples. I'll add an edit to say a little about motivation.
Apr 16, 2013 at 23:09 comment added Jim Humphreys A reference or two would help fill in background. Does the field matter? Does the reductive case involve essentially more than the semisimple case?
Apr 16, 2013 at 19:59 history asked Michael Joyce CC BY-SA 3.0