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