DIRAC COHOMOLOGY FOR COMPLEX GROUPS 19
is a symmetric pair, Helgason’s theorem implies that the K-types are of the form
(5.10), and occur with multiplicity 1.
Now we have to identify K-types E(μ) of π such that μ ρ is conjugate to τ
under W . We calculate
(5.11) μ ρ = (α1 a b + 1/2,α1 a b + 3/2,...,
αa
+ a b 3/2,αa + a b 1/2,
a b + 1/2,a b + 3/2,..., −3/2, −1/2).
b−a
To be conjugate to τ, this expression must have 2a + 1 components equal to ±1/2.
Since there is only one such component among the last b a components, the first
2a components must all be equal to ±1/2. Since the first component is smaller
than the second by one, the third component is smaller than the fourth by one,
etc., we see that the first, third etc. components must be −1/2 while the second,
fourth, etc. components must be 1/2. This completely determines μ:
(5.12) α1 = a + b 1, α2 = a + b 3,..., αa = b a + 1.
It is now clear that for this μ we indeed get a contribution to HD(π), and moreover
we can see exactly which w conjugates μ ρ to τ.
It remains to consider the case b = a. The calculation and the final result are
completely analogous. We get
(5.13) τ = (1/2, 1/2,..., 1/2),
corresponding to
(5.14) μ = (2a 1, 2a 1, 2a 3, 2a 3,..., 1, 1).
5.5. Type C. Let G = Sp(2n, C). We use the usual coordinates.
As in the other cases, the only Arthur parameters with regular integral
correspond to the principal nilpotent. In this case λ itself is integral. The only
other case when λ can be regular corresponds to the subregular
ˇ,
O corresponding
to the partition 1, 1, 2n 1. In this case, the unipotent representations are the two
metaplectic representations, πeven and πodd. The corresponding λ is given by
(5.15) = (2n 1, 2n 3,..., 3, 1).
The other cases analogous to type B are not unitary. The K-structures of πeven
and πodd are
(5.16)
(2α, 0,..., 0),
(2α + 1, 0,... 0), α N.
Here α = 0 is allowed. The WF-set is the nilpotent with columns 2n 1, 1.
Since in this case
(5.17) ρ = (n, n 1,..., 2, 1),
we see that
(5.18) τ = ρ = (n 1,n 2,..., 1, 0).
For each of the two metaplectic representations the K-types are given by μ =
(k, 0, 0,..., 0), and therefore
(5.19) μ ρ = (k n, −(n 1), −(n 2),..., −2, −1).
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