Demazure Formulas for Weight Polytopes
291
D i e
λ
=
e λ + e λ−α i + e λ−2α i + · · · + e r i λ , λ· α ∨
i 0 ;
−e λ+α i − e λ+2α i − · · · − e r i (λ+α i ) , λ· α ∨
i < 0 .
(11)
For every Weyl-group element w ∈ W a Demazure operator D w can be defined.
First, identify D r i := D i , and then use any reduced decomposition of w, replacing
the factors r j in the reduced decomposition with D j . The resulting operator D w
must be independent of which reduced decomposition is used. As a result, the
Demazure operators obey relations encoded in the Coxeter–Dynkin diagrams of X r .
For example, consider the longest element w L of the Weyl group of A 2 : w L =
r 1 r 2 r 1 = r 2 r 1 r 2 . The associated Demazure operator D w L can be written in two
ways, so that D 1 D 2 D 1 = D 2 D 1 D 2 .
If w L is the longest element of the Weyl group W , then the Demazure character
formula is
ch λ = D w L e
λ .
(12)
Also, define D i =: 1 + d i , and then d w for all w ∈ W by reduced
decompositions. Then
ch λ =
w∈W
d w e
λ .
(13)
Demazure operators can also be defined for every positive root β ∈ R + :
D(β) :=
1 − e −β r(β)
1 − e −β
,
(14)
where
r(β)λ := λ −
λ · β
∨
β and r(β)
e
λ
= e
r(β)λ .
(15)
Operators d(β) = D(β) − 1,
d(β) :=
e −β [1 − r(β)]
1 − e −β
,
(16)
are also defined for all positive roots β ∈ R + .
4 Lattice-Polytope Formulas of Demazure Type
In the hopes of helping lead to a more general result, we take a direct approach here,
and write formulas for low-rank weight-polytope sums that involve the Demazure