Let $\mathfrak{g}$ be a Kac-Moody algebra with Cartan subalgebra $\mathfrak{h}$, Weyl group $W$, and simple roots and coroots $\alpha\_i, \check{\alpha\_i}, i \in I$, respectively. Let $L$ be an integrable highest weight module. 


Write $C$ for the dominant Weyl chamber, i.e. the locus $ \{ \lambda \in \mathfrak{h}^\*: (\lambda, \check\alpha\_i ) \in \mathbb{R}^{\geqslant 0}, \forall i \in I \}$, and call the $W$ orbit of $C$ the Tits cone.


Does every weight of $L$ lie in the Tits cone?


If I haven't done something wrong, this is true for $\mathfrak{g}$ finite type and affine (untwisted). I am happy to restrict to the symmetrizable case, if that is easier to address.


Thank you in advance!



Is every weight of an integrable highest weight module in the Tits cone?
Yes. The Tits cone, as the name implies, is a cone: in particular, it's convex. Any weight for a highest weight module is an affine linear combination of finitely many extremal weights (I'll leave that as an exercise; one hint is to prove it by induction on the number of simple roots you need to get to your weight from the highest weight), that is, it's the convex hull of the extremal weights for the representation. The extremal weights are obviously in the Tits cone by definition.


