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 = e R a 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) =
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).
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