(2.122)
36 P. DEIFT, L. C. LI, AND C. TOMEI
where M{i) is real and skew.
Finally, to prove (2.117), it is sufficiently the group property for flows, to verify
(2.117) for k = 1. But for k = 1,
log(M0(A)) _ / - AMp - A 2 J 2
1 - A2
= ( 4 ^ ) ( T ^ )
by(1
-
20)-
But these factors satisfy (1.29), (1.30) and so
«7_(l,A) = ( u ;
0
- A J ) / ( l - A )
g+(l,X) = (^ + XJ)/(l + X)
by uniqueness. Thus
j - i - j ^ = M(l,X) = g+(l,X) x ( j—^5 )g+(l,X)
_
.W'Q
+ XJ. - i
,WQ
+ XJ. ,u0 AJs ,u£ + XJ
~ \ 1 + X ' ( l + X )( 1 - A ^ 1 + A
Aft (A)
1 - A 2
_ J - A M ! - A 2 J 2
1 - A 2
which completes the proof. D
The Hamiltonian H generates (2.115) on the coadjoint orbit OA" where
^ ° = ( 1 - A M
0
- A 2 J 2 ) / ( 1 - A 2 )
and
A? = (J 2 - Mo - l)/4 , A°0 = (1 + J 2 )/ 2 and A ^ = (J 2 + M
0
+ l)/ 4 (2.123)
(see Section 2(b)). By the results of the previous subsection OAQ is generic and of dimension
4N2 4N provided A J (and hence A!L\ {A\)T) is invertible and has distinct eigenvalues
(given J, this is clearly true for a dense open set of Afo's). The jTrfc's and the Arfc's of
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