6 P. DEIFT, L. C. LI, AND C. TOMEI
withS+(Mi(A)) = {A : det(c*;i - AJ) = 0}, 5_(Mi(A)) = {A : det(u^ + AJ) = 0}. Then
Af2(A) = (Wl - AJ)(cf + AJ) (1.23)
etc., factoring Mfc(A) at each stage according to the fixed spectral decomposition S =
5
+
(Mo(A))uS_(M
0
(A)) .
In the loop group approach, the simplest situation (see Section 2) arises when
5+ C {A : Re A 0} , S_ C {A : Re A 0} . (1.24)
In this case one observes that
Trr \ \ * ~
AAi
A J .
M(X) = ^ (1.25)
is a loop on the compactification S of S = iR with values in G£(N, (T), satisfying the
reality condition
M(X) = M{\) , (1.26)
and the asymptotic condition,
M(oo) is diagonal and strictly positive. (1.27)
One also observes that
" W =
(4^)(T=T (L28)
is the unique factorization of M(X) into a product of factors which are,
analytic and invertible for Re A 0, Re A 0 respectively, (1.29)
and satisfy,
("-^)L-(^)L-^-
Based on these observations we introduce the connected loop group
G = {g :g is a smooth loop from E to GC(N, (T),
contractible to the identity, and satisfying (1.26) and (1.27)}
Previous Page Next Page