Timeline for Springer Isomorphisms for Adjoint Simple Exceptional Groups
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 15, 2014 at 19:38 | comment | added | Jay Taylor | @JimHumphreys Exceptional doesn't necessarily matter but then one can't use the adjoint representation in the other types. For the classical cases, not of type $\mathrm{A}$, one could play the same game with the natural representation. It is true that there are numerous Springer isomorphism but I am looking for just one with good properties. Thus I need something I can get my hands on to show that these good properties hold. | |
| Jul 15, 2014 at 19:35 | comment | added | Jay Taylor | @PaulLevy This is somewhat sloppy. The product is being carried out in $\mathfrak{gl}(V)$. So really this should be $(k_1\mathrm{ad}(e_{\alpha_1})+1)\phi(y)$ if we remove the final $\mathrm{ad}^{-1}$ in the definition of $\phi$. | |
| Jul 15, 2014 at 19:13 | comment | added | Paul Levy | Something's not quite right - what do you mean by $(k_1e_{\alpha_1}+1)\phi(y)$? | |
| Jul 15, 2014 at 18:44 | comment | added | Jim Humphreys | Maybe it works here and even in types $G_2, F_4$, though I'm a bit skeptical about getting this simple algorithm from such arbitrary choices of root ordering. Anyway, aside from type $A_n$ the isogeny class doesn't seem to affect anything; does "exceptional" matter? Note too that there are numerous "Springer isomorphisms", considered by McNinch-Testerman: front.math.ucdavis.edu/0805.2574 | |
| Jul 15, 2014 at 13:51 | history | asked | Jay Taylor | CC BY-SA 3.0 |