8 P. DEIFT, L. C. LI, AND C. TOMEI

Remark 1.38. Note that 7r± are not projections and Ran 7r+ H Ran 7r_ ^ 0. On the

other hand Ran TT± are subalgebras of g, and hence G± = e R a n n± are subgroups of G.

This situation is similar to the classical Cholesky algorithm (see [DLT]) where one factors

a matrix M e g£(n, W),

n+M = strict upper part of M -f - diag(Af)

and

7r_M = strict lower part of M -f - diag(M) .

We show that

(1) the coadjoint orbit through Af(A), regarded as an element of #*, is finite dimen-

sional and generically of dimension AN2 — 4iV. Also, the orbit is naturally isomorphic to a

coadjoint orbit of the direct sum of two copies of the semi-direct product of (the identity

component) of G£(N,R) with g£(N,R).

(2) The Riemann-Hilbert factorization on E = iR

e

tMo(\) =g+(t,\)g_(t,\) , A G iR (1.38)

has a solution for all t e R, where g±(tr) € G± =

e

R a n 7 r ± and M0(\) = ( I - AM0 -

A2 J 2 ) / ( l — A2). By the results of i?-matrix theory,

M(t,\)=g+(t,\)-lMo(\)g+(t,\) (1.39)

solves the flow

jtM{t,\)= [(7r_logM(.-))(A) , M(t,X)] , (1-40)

and at integer times interpolates the QR-type algorithm of Moser and Veselov,

(l-A 2 )M(fc,A ) = A/*(A) , k€2S , (1.41)

(see (1.22) et seq.). Modulo technicalities (see Section 2), (1.40) is a Hamiltonian flow

which is essentially integrable (see discussion in Section 2(d)) on the (generically) 4N2 — 4N

dimensional coadjoint orbit through Afo(A).