Timeline for Reference Request: Derived group of $\mathscr R_u(B)$
Current License: CC BY-SA 3.0
7 events
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| Jun 22, 2017 at 21:01 | comment | added | Jim Humphreys | @Mikko: Yes, I've edited accordingly. I still expect to locate a more elementary proof in D_S's situation, preferably without case-by-case study of commutators or such. [It's amusing, by the way, that in his ICM Moscow 1966 talk, Steinberg himself regarded the finiteness of the number of unipotent clases to be a fairly easy problem. He had good instincts in most cases but underestimated the difficulty or was too optimistic about the correctness of some of his "problems" there.] | |
| Jun 22, 2017 at 20:59 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Jun 22, 2017 at 20:23 | comment | added | Mikko Korhonen | @JimHumphreys: I guess you know this, but regarding the last paragraph in your answer: I think that Steinberg's 1965 paper does not need finiteness of unipotent classes to prove the existence of a regular unipotent element. It takes some work, but does not need lots of machinery. | |
| Jun 22, 2017 at 18:38 | comment | added | Jim Humphreys | @Jay: What I wrote is probably too brief, but I agree that D-M have an existence proof for regular unipotents using the finiteness of the number of unipotent classes (for $p>0$). However, it's debatable whether they cover all the background for Lusztig's argument (foundations of etale cohomology, etc.). Anyway I'm still skeptical about using all this machinery to deal with the derived group of $U$, which should be more elementary. [P.S. Your reference to D-M 14.17 in a comment actually answers the question asked here apart from the method they use.] | |
| Jun 22, 2017 at 6:54 | comment | added | Jay Taylor | Jim, this is just to say that Digne--Michel do give a full proof of the existence of regular unipotent elements, at least over $\bar{\mathbb{F}}_p$. This is Theorem 14.13. They present Lusztig's argument showing that there are finitely many unipotent classes and then deduce the existence of regular unipotent elements from this. In fact, as is mentioned in the introduction to Lusztig's paper, if one knows that there are finitely many unipotent classes over $\bar{\mathbb{F}}_p$ then one knows this over any algebraically closed field $K$ of characteristic $p$. Thus their proof works over $K$. | |
| S Jun 21, 2017 at 19:33 | history | answered | Jim Humphreys | CC BY-SA 3.0 | |
| S Jun 21, 2017 at 19:33 | history | made wiki | Post Made Community Wiki by Jim Humphreys |