LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 31

under conjugation by matrices of the form g = ( *

r

J, where gr is r x r. By

differentiation, we find

tii{V\Jk(Ap),Ap}X) = 0 (2.98)

for all X of the form (

n

r ~ ). Suppose r' r. Then by the computation of Thimm

y U UN-VJ

[Th],

{Arfc,Ar-A,}(.4p) = tr(.4

p

[VA

r

r

A

(^) , V A ^ , ( ^ ) ] )

= t r ( [ ^ , V A ^

p

) ] V A £ , , ( A

p

) ) .

But as r' r, V A j ^ A p ) is clearly of the form I " r

n

)) and so {\rk, ^r'k'}(Ap) = 0 by

(2.98). This provides the remaining Ylr=i r = N{N — l)/ 2 commuting integrals.

We have proved the following result.

Theore m 2.99. The functions {Irk}irN,-r+2k2r-2 and {Arfc}ijv-i,ikr pro-

vide 2N2 — 2N commuting integrals on an orbit OA 0

We leave it to the reader to verify that on a generic leaf OA of dimension 4iV2 — 4iV,

the above integrals are independent on an open dense set. The values of the integrals for

which the gradients become dependent correspond to separatrices for the associated flows.

(d) Interpolating the Moser-Veselov algorithm

By analogy with the continuous-time Cholesky algorithm (see [DLT]), the Hamiltonian

for the flow interpolating the Moser-Veselov iteration (1.22) with splitting (1.24) is given

by

H{A)=j tv(A(z)logA(z)-A(z))^- , 4 6 ^ . (2.100)

As in the finite dimensional situation of [DLT], H is clearly not globally defined on g*. .

But there is an additional difficulty in the infinite dimensional situation: the (formal)

derivative of H is given by

dH(A) = log A = l o g ( ^ + W z ) ) .

(2-101)

which can lie in g (see discussion following (2.42)) only if Ap = 0. Thus H cannot be

differentiable in an open set. However, H is differentiable, with (directional) derivative