Demazure Formulas for Weight Polytopes
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where μ, σ is the inner product of weights μ and σ , the formal exponential e μ
simply stands for e μ,σ before a choice of weight σ is made. A choice of σ fixes
a conjugacy class of elements in the Lie group exp(X r ). The formal character then
becomes a true character ch λ (σ ), the trace, in the representation of highest weight
λ, of elements of exp(X r ) in the conjugacy class labeled by σ .
The celebrated Weyl character formula is
ch λ =
w∈W (det w) e w.λ
α∈R +
(1 − e −α )
.
(3)
Here W is the Weyl group of the simple Lie algebra X r , and w.λ = w(λ + ρ) − ρ
denotes the shifted action of Weyl-group element w ∈ W on the weight λ, with
ρ =
r
i=1 Λ i .
The Weyl invariance of the character can be made manifest:
ch λ =
w∈W
e
wλ
α∈R +
(1 − e
−wα )
−1 .
(4)
Here now
1 − e
β
−1 =
1 + e β + e 2β + . . . , β ∈ R − ;
−e −β − e −2β − . . . , β ∈ R + .
(5)
The rule-of-thumb is expand in powers of e β , with β a negative root.
2.2 Brion Lattice-Polytope Sum Formula
A polytope is the convex hull of finitely many points in R d . A polytope’s vertices are
such a set of points with minimum cardinality. A lattice polytope has all its vertices
in an integral lattice in R d . The (formal) lattice-polytope sum is the sum of terms
e φ over the lattice points φ in the polytope.
Brion [3, 4] found a general formula for these lattice-polytope sums. Let the
weight polytope Pt λ be the polytope with vertices given by the Weyl orbit W λ.
Consider the lattice-polytope sum
B λ :=
μ∈(λ+Q)∩Pt λ
e
μ ,
(6)
where the relevant lattice is the λ-shifted root lattice λ + Q of the algebra X r .
Applied to a weight polytope, the Brion formula yields