Let R be a simple
split Lie algebra over K, a field of 0 characteristic. Let π = π(λ+) be the
representation with highest weight λ+. Let Wt(λ+) be its weight system.
Let S be a subset of the root system. We define a graph grs(λ+) whose
set of nodes is Wt(λ+) and set of links is given by pairs of weights whose
difference is a root in S. In particular taking S = Σ0, the system of simple roots,
we investigate the properties of representations π(λ+) such that gr0(λ+)
is simply connected. We give a complete list of these, for each simple Lie
algebra.
We then attach to π(λ+) an affine rational lattice L(λ+), the root of λ+ in the weight
lattice modulo the root lattice and a rational polyhedron, C(λ+), the rational convex
closure of the Weyl group orbit of λ+. We give the following geometric characterization
of the weight system: Wt(λ+) = L(λ+) ∩ C(λ+)1.
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