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 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
Spin. Using the above formula for D2, we find that
Casg
+||ρg||2
CasΔ(k)
+||ρk||2
on any K-type τ occurring in Spin. Another way of putting this is
(1.1)
||χ||2
||τ +
ρk||2,
for any τ occurring in 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 = τ + ρk for some w in the Weyl group W = W (g, h).
More precisely, there is w W such that |a= 0 and |t= τ + ρk.
Conversely, if π is unitary and τ = 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
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