We leave it to the reader to verify that if K = gJg~x with deg g 0 and J in real Jordan
form, then P(K) can be taken to be the signature of the product of the nonzero, strictly
upper-triangular elements of J. In particular if K is skew, then P(K) is the sign of the
Pfaffian of A'.
If(M,A") G 0(MoJ0), then clearly A' G SK0,+ - Conversely suppose (Af, A") G S{Mo,KQ)
with K G Sx0y+, SO that K = gKog~x with deg g 0. Since tr MK£ tr MQKQ,
I = 1 , . . . , N, we have that g~xMg Mo is orthogonal, under the pairing (A, B) = tr A# ,
to the algebra generated by A'o, which is equivalent to saying that
g~lMg MQ = [to, A'o],
for some real CJ, as A'o has simple spectrum. For u = gwg~l, we then have M = gMog~l 4-
[u.glug-1]. Thus (M, K ) = Ad*igu)(M0Jo) with deg g 0.
The above calculations, together with their analogs for (R, 5), yield the following
result. We say that ((Mo, A"o), (Ro, So)) is generic if both pairs are generic.
Theore m 2.76. Suppose ((Mo, Afo),(#o, So)) is generic. Then 0(M0,A'0),(^O,SO)) a
47V2 4N dimensional manifold given by
0((A#o,*o),*o,So)) = {((M,tf),(!i,S)) : ^ MA"fc = tr M0K*, tr Kk = tr tf*,
tr RSk = tr i ^ , tr Sk = tr S£, 1 k N,
and in the case that N is even, andAo and/or
So has no real eigenvalues, then, in addition,
P(K) = P(K0) and/or P(S) = P(S0)} .
A A-i
From (2.62) we see that loops —£—H K-i0 + A\ z for which Ai and A-\ are invert-
er I z
ible with distinct eigenvalues, generate coadjoint orbits which are 4N2 4iV dimensional.
We will also call such loops generic (within the class of such loops).
A0 A0
Corollary to Theore m 2.76. Suppose A0 = —^— H —f- AQ + A\z is generic.
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