I have a simple question about root systems. Suppose that $G$ is a connected reductive group over the reals $\mathbb{R}$, and $T\subset G$ is a maximal torus (by this I mean that $T_{\mathbb{C}}$ is a maximal torus in $G_{\mathbb{C}}$). Let $R$ be the root system of $(G_{\mathbb{C}},T_{\mathbb{C}})$, and choose a set of positive roots $R^+$. Complex conjugation $\iota$ acts by Galois action on the characters of $T_{\mathbb{C}}$, and if $\chi$ is a root, then $\iota(\chi)$ is also a root.
I wonder if it's always true that if $\chi\in R^+$ then $\iota(\chi)$ is a negative root.
@dedekind: not so ill-posed, it just depends on the Cartan and set of positive roots. You are given a complex group G, a real structure $\sigma$, a $\sigma$-fixed Cartan H. Then $\sigma$ permutes the roots, and there are 3 possibilities.
Obviously real positive roots stay positive, and imaginary roots change sign, independent of the choice of positive roots. The issue is the complex roots, which can either switch sign or not, depending on the choice of positive roots. It turns out that, given a Cartan, you can always choose a set of positive roots such that $\alpha>0$ complex$\Rightarrow\sigma(\alpha)>0$ (alternatively $\alpha>0$ complex $ \Rightarrow\sigma\alpha<0$).
So the answer depends on both the Cartan and the set of positive roots. If you want to make this hold for all roots, you need to know there are no compact (resp. real) roots, which is a slightly longer story.
See Vogan's "big green book" from 1981, starting on page 4.