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46 CARINA BOYALLIAN
Here, J(ν) integral infinitesimal character if and only if
(8.1)
Again, we can assume that
and define
(8.2)
and
(8.3)
νi ∈ rZ + r ɛ
, with ɛ = 0, 1.
2
ν1 ≥ ν2 ≥ · · · ≥ |νn| ≥ 0
R(i) = Rν(i) = # j : νj = r
R(0) = Rν(0) = (2 − ɛ)#
i + ɛ
2
j : νj = r ɛ
2
, i ≥ 1
+ (1 − ɛ).
Take ν ∈ a∗ int , then J(ν) admits an invariant Hermitian form if and only
if n is even, or n is odd, ɛ = 0 and R(0) > 1. Let us see the following:
Theorem 8.4. If J(ν) has integral infinitesimal character, then an invariant
Hermitian form is positive definite on J(ν) µ , µ ∈ p, if and only if the
following conditions are satisfied:
(1) R(i + 1) ≤ R(i) + 1, for i ≥ 0;
(2) R(i + 1) = R(i) + 1, for i ≥ 1, then R(i) is odd;
(3) R(0) is odd;
(4) R(0) > 1.
Proof. See Theorem 10.3 in [BJ]. The condition R(0) > 1 does not appear
in the statement of this theorem in [BJ], but it is easy to see, checking the
proof, that, otherwise, qµ(ν) > 0.
Now, we can prove the following:
Proposition 8.5. Consider ν ∈ a∗ int such that J(ν) has integral infinitesimal
character. If conditions (1)-(4) above are satisfied then ν =
α∈Σ + cαα
with cα = 1/2 or 0.
Proof. Let us assume that r = 1, ɛ = 0. Since the Weyl group is the group
of permutation and sign changes involving an even number of signs of the
set of n elements, by condition (2) and R(0) > 1, we can suppose that νi ≥ 0
and
ν = (k, . . . , k, k − 1, . . . , k − 1, . . . . . . , 1, . . . , 1, 0, . . . , 0).
Since R(0) > 1, we have k − i ≤ n − i j=0 R(k − j), and thus
ν = 1 k−1
2
m=0
R(k−m)
r=1
k−m
[(eT (m)+r − eT (m)+r+t) + (eT (m)+r + eT (m)+r+t)],
t=1