LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 25
We will see that 0(MO,A'O) = S(M0,K0) m a n cases except the case where TV is even and K
has no real eigenvalues; in this case 0(MO,A'O) *S a n appropriate component of S(M0,A'0) a s
we show below (see Theorem 2.76).
So suppose (Mo, Ao) is generic. Define
SK0 = {K : K 1S a r e a l matrix with spec K = spec KQ} . (2.69)
By elementary linear algebra
SKo = S*0,+ U SKo,- (2.70)
where
5R-0 ± = {gKog'1 : g is a, real matrix with det g 0 ,
(2.71)
det # 0, respectively} .
There are two cases:
(1) spec A'0 DR^ 0 1
J . (2.72)
(2) spec A'
0
n-R = 0 J
(Of course, case (2) can only occur if T V is even.) In case (1), a simple compuation using
the real Jordan form (recall spec Ko is simple), shows that
SK0,+ = SA'O,- ]
(2-73)
= SK0
J
In case (2)
S/v0,+ n SA-0._ = 0 . (2.74)
This in turn follows from the observation that a matrix commuting with a real Jordan
form J = diag(I?i,... ,£JV/2) where the Bi are all 2 x 2 blocks with distinct spectrum,
must also have the same block structure and positive determinant.
In case (2), this implies that there exists a continuous map P : SK0 ~* {=^1} with the
property that for A', K' £ SK0
K = gK'g-1 , det g 0 a P(K) = P(K') . (2.75)
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