DIRAC COHOMOLOGY FOR COMPLEX GROUPS 3

In this formula, due to Parthasarathy [P1],

- Casg and Cask are the Casimir operators for g and k respectively,

- h = t + a is a fundamental θ-stable Cartan subalgebra with compatible

systems of positive roots for (g, h) and (k, t),

- ρg and ρk are the corresponding half sums of positive roots,

- Δ : k → U(g) ⊗ C(s) is given by Δ(X) = X ⊗ 1 + 1 ⊗ α(X), where α

is the action map k → so(s) followed by the usual identiﬁcations so(s)

∼

=

2

(s) → C(s).

If π is unitary, then Hπ ⊗ Spin admits a K-invariant inner product , such that

D is self-adjoint with respect to this inner product. It follows that D2 ≥ 0 on

Hπ ⊗ Spin. Using the above formula for D2, we ﬁnd that

Casg

+||ρg||2

≤ CasΔ(k)

+||ρk||2

on any K-type τ occurring in Hπ ⊗ Spin. Another way of putting this is

(1.1)

||χ||2

≤ ||τ +

ρk||2,

for any τ occurring in Hπ ⊗ Spin, where χ is the inﬁnitesimal character of π. This

is the Dirac inequality mentioned above.

These ideas are generalized by Vogan [V2] and Huang-Pandˇ zi´ c [HP1] as fol-

lows. For an arbitrary admissible (g,K)-module π, we deﬁne the Dirac cohomology

of π as

HD(π) = ker D/(ker D ∩ im D).

Then HD(π) is a module for K. If π is unitary, HD(π) = ker D = ker D2.

The main result about HD is the following theorem conjectured by Vogan.

Theorem 1.2. [HP1] Assume that HD(π) is nonzero, and contains an ir-

reducible K-module with highest weight τ. Let χ ∈ h∗ denote the inﬁnitesimal

character of π. Then wχ = τ + ρk for some w in the Weyl group W = W (g, h).

More precisely, there is w ∈ W such that wχ |a= 0 and wχ |t= τ + ρk.

Conversely, if π is unitary and τ = wχ − ρk

is the highest weight of a K-type

occurring in π ⊗ Spin, then this K-type is contained in HD(π).

This result might suggest that diﬃculties should arise in passing between K-

types of π and K-types of π ⊗ Spin. For unitary π, the situation is however greatly

simpliﬁed by the Dirac inequality. Namely, together with (1.1), Theorem 1.2 shows

that the inﬁnitesimal characters τ+ρk of K-types in Dirac cohomology have minimal

possible norm. This means that whenever such E(τ) appears in the tensor product

of a K-type E(μ) of π and a K-type E(σ) of Spin, it necessarily appears as the

PRV component [PRV], i.e.,

(1.2) τ = μ +

σ−

up to Wk,

where σ− denotes the lowest weight of E(σ).

For unitary representations, the relation of Dirac cohomology to (g,K) coho-

mology is as follows. (For more details, see [HP1] and [HKP].) One can write the

K-module (s) as Spin ⊗ Spin if dim s is even, or the direct sum of two copies of

the same space if dim s is odd. It follows that

HomK ( (s),π ⊗ F

∗)

= HomK (F ⊗ Spin, π ⊗ Spin),

3

In this formula, due to Parthasarathy [P1],

- Casg and Cask are the Casimir operators for g and k respectively,

- h = t + a is a fundamental θ-stable Cartan subalgebra with compatible

systems of positive roots for (g, h) and (k, t),

- ρg and ρk are the corresponding half sums of positive roots,

- Δ : k → U(g) ⊗ C(s) is given by Δ(X) = X ⊗ 1 + 1 ⊗ α(X), where α

is the action map k → so(s) followed by the usual identiﬁcations so(s)

∼

=

2

(s) → C(s).

If π is unitary, then Hπ ⊗ Spin admits a K-invariant inner product , such that

D is self-adjoint with respect to this inner product. It follows that D2 ≥ 0 on

Hπ ⊗ Spin. Using the above formula for D2, we ﬁnd that

Casg

+||ρg||2

≤ CasΔ(k)

+||ρk||2

on any K-type τ occurring in Hπ ⊗ Spin. Another way of putting this is

(1.1)

||χ||2

≤ ||τ +

ρk||2,

for any τ occurring in Hπ ⊗ Spin, where χ is the inﬁnitesimal character of π. This

is the Dirac inequality mentioned above.

These ideas are generalized by Vogan [V2] and Huang-Pandˇ zi´ c [HP1] as fol-

lows. For an arbitrary admissible (g,K)-module π, we deﬁne the Dirac cohomology

of π as

HD(π) = ker D/(ker D ∩ im D).

Then HD(π) is a module for K. If π is unitary, HD(π) = ker D = ker D2.

The main result about HD is the following theorem conjectured by Vogan.

Theorem 1.2. [HP1] Assume that HD(π) is nonzero, and contains an ir-

reducible K-module with highest weight τ. Let χ ∈ h∗ denote the inﬁnitesimal

character of π. Then wχ = τ + ρk for some w in the Weyl group W = W (g, h).

More precisely, there is w ∈ W such that wχ |a= 0 and wχ |t= τ + ρk.

Conversely, if π is unitary and τ = wχ − ρk

is the highest weight of a K-type

occurring in π ⊗ Spin, then this K-type is contained in HD(π).

This result might suggest that diﬃculties should arise in passing between K-

types of π and K-types of π ⊗ Spin. For unitary π, the situation is however greatly

simpliﬁed by the Dirac inequality. Namely, together with (1.1), Theorem 1.2 shows

that the inﬁnitesimal characters τ+ρk of K-types in Dirac cohomology have minimal

possible norm. This means that whenever such E(τ) appears in the tensor product

of a K-type E(μ) of π and a K-type E(σ) of Spin, it necessarily appears as the

PRV component [PRV], i.e.,

(1.2) τ = μ +

σ−

up to Wk,

where σ− denotes the lowest weight of E(σ).

For unitary representations, the relation of Dirac cohomology to (g,K) coho-

mology is as follows. (For more details, see [HP1] and [HKP].) One can write the

K-module (s) as Spin ⊗ Spin if dim s is even, or the direct sum of two copies of

the same space if dim s is odd. It follows that

HomK ( (s),π ⊗ F

∗)

= HomK (F ⊗ Spin, π ⊗ Spin),

3