For printing - MSP

38 CARINA BOYALLIAN
where k ≤ [ n
2 ] − 1.
In order to complete this proof, we will need the following:
Lemma 3.4. Take ν as above and consider R(0) ≥ 1 (otherwise, ν ≡ 0 by
(1)). Then
⎛
⎞
(3.5)
i
2(k − i) − 2 < n − ⎝2 R(k − j) ⎠ + 1 where i = 0, . . . , k − 1.
j=0
Proof. Since n = 2 k−1 j=0 R(k − j) + R(0) we have
2(k − i) − 2 ≤ 2
k−1
j=i+1
Then, (3.5) has been proved.
R(k − j) + R(0) = n − 2
< n − 2
i
R(k − j)
j=0
i
R(k − j) + 1.
j=0
Proof of Proposition 3.2 (Continuation). Let define
m−1 t=0 R(k − t),
T (m) =
0,
if m > 0
if m = 0 .
(3.6)
Now, it is easy to check, that Lemma 3.4 allows us to rewrite ν as follows:
ν = 1
k−1
R(k−m)
(eT (m)+j − en−T 2
(m)−j+1)
m=0 j=1
2(k−m)−2+T (m)+j
+
[(eT (m)+j − es+1) + (es+1 − en−T (m)−j+1)]
s=T (m)+j
And so, Proposition 3.2 is proved.
Proof of Theorem 2.1 for Sl(n, F) with F = R, C. This is immediate from
Proposition 3.2, Theorem 3.1 and Proposition 1.5.
4. Case Sp(2n, F), n > 2 with F = R, C.
Now, assume that G = Sp(2n, R) or G = Sp(2n, C). In this case, Σ(g0, a0)
is of type Cn, and identifying a C n we have that the restricted roots are
{±ei ± ej, ±2el}. Put r = dimR F.
Here, J G (ν) has integral infinitesimal character if and only if νi ∈ rZ.
Take ν ∈ a ∗ , and replace it by a Weyl group conjugate, then we may assume
.