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 2λ 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) 2λ = (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 = 2λ − 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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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 2λ 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) 2λ = (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 = 2λ − 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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