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M. A. Walton
B λ =
w∈W
e
wλ
α∈S
(1 − e
−wα )
−1 .
(7)
Here S denotes the set of simple roots of X r .
2.3 Polytope Expansion
The Brion formula (7) is remarkably similar to the Weyl character formula, as
written in (4) [6, 15]. It is therefore natural, and fruitful, to consider the polytope
expansion of Lie characters [6, 15, 16]:
ch λ =
μ≤λ
polyt λ (μ) B μ .
(8)
The polytope multiplicities polyt λ (μ) are defined in analogy with weight multiplicities mult λ (μ).
We do not consider the polytope expansion further in this note. Instead we focus
on the striking relation described above between characters and polytope sums.
3 Demazure Character Formulas
Do other character formulas point to the existence of new formulas for the lattice
sums of weight polytopes? More general polytopes?
In particular, do the Demazure formulas for Lie characters indicate the existence
of Demazure-type formulas for the lattice sums of weight polytopes?
Let us first sketch the Demazure character formulas. The Weyl group W is
generated by the primitive reflections r i in weight space across the hyperplanes
normal to the corresponding simple roots α i :
r i λ = λ −
λ · α
∨
i
α i ;
(9)
here α ∨
i = 2α i /(α i · α i ) is the simple co-root.
Define Demazure operators for every simple root α i ∈ S:
D α i =: D i =
1 − e −α i r i
1 − e −α i
,
(10)
where r i (e λ ) = e r i λ . Then