Timeline for Haar measure on strictly upper triangular matrices
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2010 at 23:51 | comment | added | BCnrd | @Pete: the wound stuff is not so nasty as you may think. See Appendix B of the pseudo-reductive book. Also see Oesterle's paper; he proves over non-arch local field they have compact group of local pts, just like an anisotropic torus. | |
| Apr 15, 2010 at 19:25 | comment | added | Pete L. Clark | @Brian: right, there's this "k-wound" nastiness that I read about in BLR. (That's why I said e.g.) Thanks for your response. | |
| Apr 15, 2010 at 19:22 | comment | added | BCnrd | @Pete: Even Witt groups have filtration by copies of the additive group, so there are far worse unipotent groups out there (over imperfect fields). That said, they don't arise for the question as posed, for the reason mentioned in your first comment and mine (I have the bad habit to say "Hilbert 90" for the additive version too, which I suppose must have another name.) | |
| Apr 15, 2010 at 18:33 | comment | added | Pete L. Clark | @Brian: I know that in general there are more unipotent groups in positive characteristic, e.g. Witt vector groups. But do they arise in this case? | |
| Apr 15, 2010 at 18:31 | comment | added | BCnrd | If $G$ is any solvable smooth connected affine gp over global $F$ and has no nontrivial $F$-rat'l character (e.g., unip.) then $G(A)/G(F)$ is compact. Over number fields, due to Godement. Argument generalized to function fields by Oesterle; see Thm 1.3 in Ch. IV of Oesterle's Inv. paper on Tamagawa numbers. For case you ask about, just use comp. series with successive qts as vector gps, and lots of Hilb. 90 for exactness on adelic, local, global pts. In fact, for any smooth conn'd affine $G$ over $F$ with no nontriv $F$-rat'l characters, $G(A)/G(F)$ has finite vol; lies deeper in char > 0. | |
| Apr 15, 2010 at 18:23 | comment | added | Pete L. Clark | Isn't such a group a repeated extension of copies of the adele class group (which is compact)? | |
| Apr 15, 2010 at 17:32 | history | asked | Phil | CC BY-SA 2.5 |