20 DAN BARBASCH AND PAVLE
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This should be equal to τ up to W , and this happens precisely when k = n. (Recall
that W consists of permutations and arbitrary sign changes.)
So we see that for even n, HD(πeven) consists of E(τ), for τ as in (5.18), without
multiplicity other than the global multiplicity [Spin : E(ρ)], while HD(πodd) = 0.
For odd n, the situation is reversed: HD(πeven) = 0, while HD(πodd) consists
of E(τ), with multiplicity [Spin : E(ρ)].
5.6. Type D. Let G = SO(2n, C). We use the usual coordinates.
Since must be regular integral, in this case the WF-sets of the nontrivial
unipotent representations can only be nilpotents with columns 2b, 2a 1, 1, where
a + b = n.
By [B], there are two unipotent (so also unitary) representations with W -
conjugate to (2a−1, 2a−3,... , 1; 2b−2,..., 0); the spherical one, and the one with
lowest K-type (1, 0,..., 0) and parameter
(a 1/2,..., 3/2, −1/2,b 1,..., 1, 0) × (a 1/2,..., 3/2, 1/2,a 1,..., 1, 0).
Made dominant for the standard positive system,
(5.20) = (2b 2, 2b,..., 2a + 2, 2a, 2a 1, 2a 2,..., 1, 0).
(When b = a, the parameter is (2a 1, 2a 2,..., 1, 0).) Since
(5.21) ρ = (a + b 1,a + b 2,..., 1, 0),
we see that
(5.22) τ = ρ = (b a 1,..., 1, 0, 0,..., 0)
2a
The K-structure of our unipotent representations is given by
(5.23)
μ = (α1,...,α2a, 0,..., 0), αj N, αj 2N,
μ = (α1,...,α2a, 0,..., 0), αj N, αj 2N + 1.
The argument is similar to type B, but more involved because π is not trivial.
The first case is for the spherical representation, the second for the other one.
Therefore,
(5.24) μ ρ = (α1 (a + b 1),...,α2a (b a), −(b a 1),..., −1, 0).
Since τ has 2a + 1 zeros, the only way μ ρ can be conjugate to τ is to have
(5.25) α1 = a + b 1, α2 = a + b 2,...,α2a = b a.
Using (5.23), we conclude that for even a, the spherical unipotent representation
has HD equal to E(τ), with multiplicity [Spin : E(ρ)], while the nonspherical
representation has HD = 0. For odd a the situation is reversed: the spherical
representation has HD = 0, while the nonspherical one has HD equal to E(τ),
with multiplicity [Spin : E(ρ)]. (Recall that for type D the Weyl group consists of
permutations combined with an even number of sign changes, but we can use all
sign changes because we are wroking with the full orthogonal group.)
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