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)
(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
on any K-type τ occurring in Hπ ⊗ Spin. Another way of putting this is
≤ ||τ +
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 +
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),