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)}