Timeline for Characters separating points on Maximal Torus modulo Weyl group?

5 events

when toggle format	what		by	license	comment
Jan 12, 2013 at 18:36	answer	added	Jim Humphreys		timeline score: 4
Jan 12, 2013 at 18:08	comment	added	Jeep Wrangler		Many thanks to your comment. Now I feel more secure. As you may know, by Stone-Weierstrass theorem, this would imply that the linear combination of characters would be dense in the space of continuous function on T/W. It is something like Peter-Weyl theorem which says that all continuous functions can be approximated by linear combinations of characters in L^2 norm.
Jan 12, 2013 at 17:53	comment	added	user29720		I should have written $Z_G(t')^0$ rather than $Z_G(t)^0$ above.
Jan 12, 2013 at 17:37	comment	added	user29720		Presumably $G$ is connected. Points $t, t' \in T$ have distinct images in $T/W$ iff they are not conjugate in $G$. Indeed, if $t' = gtg^{-1}$ then $gTg^{-1}$ and $T$ contain $t'$ and so lie in the connected compact Lie group $Z_G(t)^0$. By conjugacy of maximal tori in such Lie groups, there exists $h \in Z_G(t)^0$ such that $h(gTg^{-1}) = T$, so $t' = ht'h^{-1} = (hg)t(hg)^{-1}$ with $hg \in N_G(T)$, so $t'$ is in the $W$-orbit of $t$. So your question is exactly if characters of $G$ separate conjugacy classes in $G$ (as every $g \in G$ lies in a torus, since $G$ is connected and compact).
Jan 12, 2013 at 16:56	history	asked	Jeep Wrangler	CC BY-SA 3.0