4 P. DEIFT, L. C. LI, AND C. TOMEI
(see [V]),
Mfc+i =u;kMkujZ , (1.12)
where
Mk = u%J - Jw& £ o(N) . (1.13)
In the variables (a;*, M*), the nature of the above correspondence is refelcted in the fact
that u*k is not uniquely determined by Mk through (1.13). The choice of a particular
mapping reduces, given Mk, to a particular choice of matrix uk £ 0(N) in (1.13).
Moser and Veselov proceed as follows. They consider the closed 2-form
u = tr dXJ A dYT = ^ J, dXij A d*-,- (1.14)
restricted to Q2n = O(iV) x O(N). A straightforward, but somewhat tedious, computation
shows that,
oj is nondegenerate at (X,Y) £ O(N) x O(N)
& (1.15)
A + A' ^ 0 for all A, A' £ s p e c ( r T X J " 1 ) .
In other words, a; is nondegenerate at a point (Xk-i,Xk) £ Q2n only if A -f A' ^ 0
for all (generalized) eigenvalues A, A', det(ujt A J ) = det(u;jfc A'J) = 0, where again
ujk = X^Xk-i- The basic observation in [MV] is that (1.13) is equivalent to the matrix
polynomial factorization
Mk(\) = I - AM* - A2 J 2 = (uJ + \J)(«k - A J ) (1.16)
and switching factors
M*+1(A) = (u;*-AJ)(u;jf + AJ)
(1.17)
=
I-\Mk+i-\2J2
yields Mk+i = u;fcMfcu;jf, by (1.12). For (Xk-i^Xk) satisfying (1.15), the above factoriza-
tion for Mfc(A) has the property that for
S = {A : det( J - AAfjb - A2 J 2 ) = 0} ,
S+ = {A : det(u;* - AJ) = 0} , (1.18)
S- = {A:det(wjf+ AJ) = 0} ,
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