28 P. DEIFT, L. C. LI, AND C. TOMEI
for all r, fc 2Z .
T h e o r e m 2.80. The /rfc's are a family of smooth, Poisson commuting functions on
g*. . The derivative dlrk is given by
dIrk(A) = (z- l)z~k j _ tv[adj((z - l)A(z) - v ) } ~ ^ £ , (2-81)
where adj(-) denotes the classical adjoint matrix.
Proof: The functions are clearly smooth (with respect to any reasonable topology on
g*. ) and the proof of (2.81) is a direct computation. By the results of f2-matrix theory
(see Appendix), to prove commutivity it is enough to show that the 7rfc's are invariant
under the Ad* action of G,
Irk(k?gA) = Irk(A) (2.82)
for all g G, where
(Ad;A)(z) = g-\z)A{z)g{z) £*ng . (2.83)
But this follows trivially from the invariance of the determinant. D
Notice in particular from (2.81),
dIrk(A)(z = 1) = 0 . (2.84)
By general i2-matrix theory, the Jrjt's generate flows of the form
A = [ic-dIrk(A),A] (2.85)
on generic coadjoint orbits 0A = 0Ap/(z-.1)+A_lZ-i+Ao+AlZ C £*
ng
. Thus
Ap = [(7r-dIr*(A))(z), (z - D-4]|
r = 1
= 0 , (2.86)
by (2.84). Thus the entries of Ap provide additional integrals for the Irk flows.
On generic orbits 0A most of the /rjfc's degenerate: we compute the nontrivial intgrals
that remain. For (z - l)A(z) = Ap + (z - l)(.4_ic _ 1 + A0 + A1z),
N
det((z-l)A(z)-ri) = Y,Er(-z-1A-1^(Ap^A.l-A0)+z(A0-A1)WA1)7]n-r, (2.87)
r=0
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