DIRAC COHOMOLOGY FOR COMPLEX GROUPS 15
Langlands parameter given by a minimal principal series of G. The Levi compo-
nent is M0 = U(1)b × U(a b). This principal series has to contain μ. But μ is
1-dimensional, and its restriction to M0 is clear. The result follows from computing
the parameter of a principal series whose Langlands subquotient is finite dimen-
sional and contains μ. The multiplicity follows from the fact that μ occurs with
multiplicity 1.
Recall that we need to consider the K-type with highest weight τ = ρ and
try to realize it in the tensor product π Spin, or equivalently in π E(ρ), as a
PRV component. Since the number of coordinates is a + b,
(4.3) ρ =
α + β
2
+
k + l
2
, . . . ,
α + β
2

k + l
2
.
It follows that τ equals
(4.4)
τ =
β + α
2
+
k l
2
+ k, . . . ,
β + α
2
+
k l
2
+ 1,
β + α
2
+
k l
2
, . . . ,
β + α
2
+
k l
2
,
β + α
2
+
k l
2
1,...,
β + α
2
+
k l
2
l
On the other hand, since the K-types of π have highest weight equal to the sum of
ξ and roots in Δ(n), μ ρ has coordinates
(4.5)
μ ρ =
+ 1
2
+
k l
2
+ x1,...,
β + α
2
+
k l
2
+ l + xb,
β + α
2
+
k l
2
+ l + 1 + xb+1,...,
β + α
2
+
k l
2
+ k + xa,
β + α
2
+
k l
2
l yb,...,
1
2
+
k l
2
y1
with
(4.6)
· · · xi xi+1 · · · 0
· · · yj yj+1 · · · 0.
The coordinates in the middle of μ−ρ are term by term bigger than the coordinates
appear at the beginning of τ. This forces xa−k+1 = · · · = xa = 0. By the same
argument yb = · · · = yb−l+1 = 0. Note from formula 4.2 that a k and b l.
The coordinates that are left over from τ are all equal, so the remaining yj,xi are
uniquely determined.
5. Unipotent representations with Dirac cohomology
In this section we give an exposition of unipotent representations, and compute
Dirac cohomology for many examples.
5.1. Langlands Homomorphisms. In order to explain the parameters of
unipotent representations we recast the classification of (g,K)-modules in terms of
Langlands homomorphisms.
First some notation: For the field of reals the Weil group is
WR :=

· {1,j},
j2
= −1
C×, jzj−1
= z,
15
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