integral. Replace w1λ by λ. Thus we can write the parameter of π as (λ, −sλ) with
λ dominant, and s W. Since L(λ, −sλ) is assumed unitary, it is Hermitian. So
there is w W such that
(2.4) w(λ + sλ) = λ + sλ, w(λ sλ) = −λ + sλ.
This implies that = sλ, so w = s since λ is regular, and wsλ = s2λ = λ. So s
must be an involution.
Thus to compute HD(π) for π that are unitary, we need to know
(1) L(λ, −sλ) that are unitary with
(2.5) = τ + ρ,
in particular is regular integral,
(2) the multiplicity
(2.6) L(λ, −sλ) E(ρ) : E(τ) .
2.2. Unitarily induced representations. We consider the Dirac cohomol-
ogy of a representation π which is unitarily induced from a unitary representation
of the Levi component M of a parabolic subgroup P = MN.
We write π := IndP
[Cξ πm], where ξ is a unitary character of M, and πm
is a unitary representation of M such that the center of M acts trivially. It is
straightforward that πm has Dirac cohomology if and only if πm has Dirac
The representation πm = Lm(λm, −sλm) satisfies
λm + sλm = μm, 2λm = μm + νm, (2.7)
λm sλm = νm, 2sλm = μm νm, (2.8)
with s Wm. Assume that πm has Dirac cohomology. So
(2.9) 2λm = μm + νm = τm + ρm
is regular integral and dominant for a positive system Δm. Here τm is dominant
with respect to Δm, and ρm is the half sum of the roots in Δm. Also,
(2.10) πm F (ρm) : F (τm) = 0.
Notation 2.3. For a dominant m-weight χ, we denote by F (χ) the finite-
dimensional m-module with highest weight χ. For a dominant g-weight η, we denote
by E(η) the finite-dimensional g-module with highest weight η. We are also going
to use analogous notation when χ and η are not necessarily dominant, but any
extremal weights of the corresponding modules.
The lowest K-type subquotient of π is L(λ, −sλ). It has parameters
λ = ξ/2 + λm, μ = ξ + μm,
= ξ/2 + sλm, ν = νm.
We assume that ξ is dominant for Δ(n) the roots of N. This is justified in view of
the results in [V1] and [B] which say that any unitary representation is unitarily
induced irreducible from a representation πm on a Levi component with these prop-
erties. In order to have Dirac cohomology, must be regular integral; so assume
this is the case. Let Δ be the positive system such that λ is dominant. Then
(2.12) = ξ + μm + νm = τ + ρ .
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