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 + sλ = ξ, 2λ = ξ + 2ρm, (3.3)
λ − sλ = 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 2λ is dominant. Then 2λ = ρ and
τ = 0. Thus
ξ = ρ − 2ρm, sρ = ρ − 4ρm.
Next, the formula (2.13) for the case of general λ simpliﬁes to
π : E(ρ) = Cξ : E(ρ)|m = Cξ : F (ρm) ⊗ C−ρn ⊗
Cξ ⊗ F (ρm) ⊗ Cρn :
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
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
ment of B in Δ. Then
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
= 2ρ(B), 2ρ − 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 = (Δ ∩ wΔ) ∪ (Δ ∩ −wΔ), and that
consequently ρ + wρ = 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