LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 37
Theorem 2.49 provide 2JV2 2N commuting integrals for H. Indeed, {Irk,H}9* = 0 by
Ad*-invariance and i?-matrix theory: alternatively from (2.114),
Irk(M(t,-)) = Irkig+it^O)-1 M(t, •)?+(*, 0))
= I
r t
( M ( 0 , - ) )
On the other hand, the conservation of the Arfc's is trivial. The integrals 7rfc, Arfc, however,
are clearly dependent on the invariant set
MAo = {A £ 0Ao : Irk(A) = Irk(A°) , l r i V , - r + 2 f c 2 r - 2 ,
Xrk(A) = Xrk(A°) = 0 , l r J V - l , l i f c r }
and so the if-flow with the relevant initial conditions always lies on a separatrix of the
integrable scheme. The flow on the separatrix can be analyzed by noting that H induces
a flow t —• M(t) on the reduced space o(N)*. As we now show, and as anticipated from
[MV], this flow is completely integrable in the classical sense.
The equation of motion for M(t) is obtained by differentiating (2.115) with respect
to A and setting A = 0. We find
^
=
[ ( , . l o
g
M ( V ) ) ( 0 ) , M
W
] ,
( 2 i 2 4 )
M(0) = Mo .
From (2.114) we have
I - \M(t) - A2 J 2 = I -Xg+it^r1 [Mo + A J 2 ] g+(t,\)
and letting A 0,
M(t) =g+(t,0)-1.Mog+(t,0) . (2.125)
But more is true: M(t) = —M(t)T is orthogonally equivalent to Afo, as it should be.
Indeed if
0+ft,A) = |0+(*,A|Q+(,A) , XeiR (2.126)
is the polar decomposition of g+(t. A), then using (2.111) in (2.110), we obtain
\g+(t,X)\ = eit/2)lo*(i"o(x)) (2.127)
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