6 DAN BARBASCH AND PAVLE

PANDˇ

ZI

´

C

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 wλ = 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) 2λ = τ + ρ,

in particular 2λ 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

G

[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 Cξ ⊗ πm has Dirac

cohomology.

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

(2.11)

λ = ξ/2 + λm, μ = ξ + μm,

sλ = 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, 2λ must be regular integral; so assume

this is the case. Let Δ be the positive system such that λ is dominant. Then

(2.12) 2λ = ξ + μm + νm = τ + ρ .

6

PANDˇ

ZI

´

C

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 wλ = 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) 2λ = τ + ρ,

in particular 2λ 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

G

[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 Cξ ⊗ πm has Dirac

cohomology.

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

(2.11)

λ = ξ/2 + λm, μ = ξ + μm,

sλ = 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, 2λ must be regular integral; so assume

this is the case. Let Δ be the positive system such that λ is dominant. Then

(2.12) 2λ = ξ + μm + νm = τ + ρ .

6