Let $G$ be a split reductive group and let $T$ be a split maximal torus whose rank is $l$. Is it possible to find a base $\gamma\_1,..., \gamma\_l$ of the weight lattice $X(T)$ such that the cone $C$ in the coweight lattice $Y(T)$ defined by
$$C=\{\psi\in Y(T)\vert (\psi, \gamma\_i)\geq 0\; i=1,...,l\}$$
forms a complete fan under the translation of the Weyl group in $Y(T)$?


If $G$ is an adjoint group, such a base is provided by simple roots and $C$ is simply the positive Weyl chamber. How about the general reductive group?



Find an analogue of Weyl chamber structure
Saying that the translates of C form a complete fan seems to imply that $C$ is a fundamental domain for the Weyl group: The translates cover space, and they form a fan, meaning the intersection of any two translates has lower dimension.


The Weyl group is generated by reflections. The interior of a fundamental domain can't intersect the fixed hyperplanes of these reflections, because then the intersection of the domain with its translate by that reflection must have full dimension. Since the fundamental domain is a cone, its interior is connected, and thus lies in a single connected component of the complement of these hyperplanes, i.e. a single Weyl chamber. Since its translates cover the space, it must make up a full Weyl chamber. So $C$ must be a Weyl chamber.


After translating, we can assume $C$ is the standard Weyl chamber. This happens only if the $\gamma\_i$s are positive real multiples of the simple roots.


So this is possible if and only if the least positive multiples of the simple roots that lie in the weight lattice in fact form a basis for the weight lattice. This fails, for example, for $SL\_n$.


