Let $\mathfrak{g}(A)$ be the Kac-Moody Lie algebra associated to an indecomposable generalized Cartan matrix $A$, and let $Z$ be the convex hull in $\mathfrak{h}^*_{\mathbb{R}}$ of the set $\Delta_+^{im}\cup \{0\}$. Then is the set of rays through $\Delta_+^{im}$ dense in $Z$?
Some context; if the answer is yes this seems like a pretty useful statement, and I came across this statement in Kac's book, proof of prop 5.9, page 66 (3rd edition), stating that every root basis is Weyl conjugate to the standard one $\Pi$, or $-\Pi$. There, the statement is claimed to follow from the following characterization of the imaginary roots: $\alpha$ is an imaginary root if there is an element in the Weyl group $w$ such that $\langle w(\alpha),\alpha^{\vee}_i\rangle\leq 0$ for all $i$, and the support of $w(\alpha)$ is connected.
Some further context: where this statement appears in the proof of Prop 5.9 in Kac's book, it seems that we could get by with the weaker statement:
"Let $S$ be the closure of the rays through positive imaginary roots, then the intersection of $S$ with the unit sphere is connected."
This actually does seem to follow from the characterization of imaginary roots given above. Is the answer to the question in the title perhaps no?
As it is, it's hard to see how this characterization of the imaginary roots helps with the question in the title. For instance take two imaginary roots $\alpha^{s}$, for $s=1,2$ such that $\langle \alpha^s,\alpha^{\vee}_i\rangle\leq 0$ for all $i$, and the supports of $\alpha^1$ and $\alpha^2$ are disjoint and disconnected. This can of course still happen even if $A$ is indecomposable. Then in the convex cone there will be a bunch of elements like $t\alpha^1+(1-t)\alpha^2$, with disconnected support, that we are somehow meant to approximate with imaginary roots with connected support via the action of the Weyl group. I guess, up to the action of the Weyl group we can assume that $\alpha^1$ has full support, since $A$ is indecomposable, but I don't see how it helps.
Has anyone any idea what's going on here, or a reference apart from Kac's book?
