10 DAN BARBASCH AND PAVLE
PANDˇ
ZI
´
C
3.1. Induced from a unitary character, infinitesimal character ρ/2.
We look at the special case when π has infinitesimal character ρ/2, and is unitarily
induced from a unitary character ξ on a Levi component m. In this case we will be
able to improve over the result of 2.2.
Choose a positive system Δ so that ξ is dominant, and let p = m + n be the
parabolic subalgebra determined by ξ. The representation π = L(λ, −sλ) satisfies
λ + = ξ, = ξ + 2ρm, (3.3)
λ = 2ρm, 2sλ = ξ 2ρm. (3.4)
It can be shown that this implies s = wm, the long Weyl group element in W (m).
This fact is however not needed in the following.
Let Δ be a positive root system so that is dominant. Then = ρ and
τ = 0. Thus
ξ = ρ 2ρm, = ρ 4ρm.
Next, the formula (2.13) for the case of general λ simplifies to
(3.5)
π : E(ρ) = : E(ρ)|m = : F (ρm) C−ρn

n =
F (ρm) Cρn :

n .
The LHS of the last line of (3.5) has highest weight
(3.6) ξ + ρm + ρn = ρ 2ρm + ρm + ρn = ρ + wmρ,
and lowest weight
(3.7) wm(ρ + wmρ) = wmρ + ρ.
We have already shown in Subsection 2.2 that (3.5) is nonzero; we now show that
it is equal to 1, i.e., that the multiplicity of F (wmρ + ρ) in

n is equal to 1. We
are going to use some classical results of Kostant [K1], [K2] which we describe in
the following.
If B Δ, denote by 2ρ(B) the sum of roots in B. In this notation,
(3.8) ρ + wmρ = 2ρ(B),
where B := α Δ(n) : ρ , α 0 .
Lemma 3.1 (Kostant). Let B Δ be arbitrary, and denote by
Bc
the comple-
ment of B in Δ. Then
2ρ(B),
2ρ(Bc)
0,
with equality if and only if there is w W such that 2ρ(B) = ρ + wρ. In that case,
B is uniquely determined by w as B = Δ wΔ.
Proof. Since 2ρ(B) + 2ρ(Bc) = 2ρ, we have
(3.9) 2ρ(B),
2ρ(Bc)
= 2ρ(B), 2ρ(B) = ρ, ρ ρ 2ρ(B),ρ 2ρ(B) .
But ρ 2ρ(B) is a weight of E(ρ), so the expression in (3.9) is indeed 0. It is
equal to 0 precisely when ρ−2ρ(B) is an extremal weight of E(ρ). In that case it is
conjugate to the lowest weight −ρ, i.e., there is w W such that ρ−2ρ(B) = −wρ.
For the last statement, notice that Δ = wΔ) −wΔ), and that
consequently ρ + = 2ρ(Δ wΔ), since the elements of Δ −wΔ cancel out in
the sum ρ + wρ.
Corollary 3.2. The weight ρ + wmρ occurs with multiplicity 1 in

n.
10
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