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)