Timeline for centralisers of maximal split tori

7 events

when toggle format	what		by	license	comment
Dec 4, 2013 at 8:42	vote	accept	Rupert
Dec 3, 2013 at 16:38	answer	added	Jim Humphreys		timeline score: 1
Dec 3, 2013 at 16:16	comment	added	Jim Humphreys		@Rupert: I added a more relevant tag.
Dec 3, 2013 at 16:15	history	edited	Jim Humphreys		edited tags
Dec 3, 2013 at 14:53	comment	added	user76758		I think there are also some E$_6$ counterexamples as well (over suitable fields). For "most" types it is OK by inspection of classification theorems.
Dec 3, 2013 at 14:50	comment	added	user76758		No. Assume $G$ is semisimple (else silly counterexamples via non-split central tori). For a maximal $k$-torus $T\supset S$ and basis $\Delta$ of the set $\Phi$ of absolute roots, a basis of absolute roots of $Z_G(S)$ is $\Delta_0=\ker(\Delta\rightarrow{\rm{X}}(S))$. You want $\Delta_0$ of size $\dim(T)-\dim(S)$; i.e., $D:=\Delta -\Delta_0$ of size $\dim(S)$ $(\dim(T)=\#\Delta$ since $G$ is semisimple). Restriction $D\rightarrow{\rm{X}}(S)$ maps onto a basis of relative roots, so you want it injective; i.e., $\#D$ is the $k$-rank. False for many non-quasi-split unitary groups over fields.
Dec 3, 2013 at 12:45	history	asked	Rupert	CC BY-SA 3.0