LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 33

provides the (unique) factorization of etX^x\

etxw = g+(t, A) (?_(t, A) , g± G G . (2.108)

with g±(t, •) analytic in {Re A 0}, {Re A 0} respectively, continuous and invertible in

{Re A 0 } , { R e A 0 } respectively, and satisfying (2.102). (Note that the requirement

g± £ G implies, in particular, that g±(t, oo) 0, which is needed for uniqueness.)

Now suppose that (X-\, -Yo) is a pair in Q2n = O(N) x O(N) for which the associated

matrix UJQ = XjX-\ has spectrum S± satisfying (1.24). Then the associated loop

1 - AM0 - A2 J 2 _ (u;0T + A J)(uQ - A J )

M0(A) =

1 - A2 1 - A2

is invertible on E. Moreover, as the loop is clearly self-adjoint on E, with limit J2 0 at

infinity, the loop is necessarily (strictly) positive definite. It follows that

.Y

0

(A)=log(Mo(A)) (2.109)

(take the principal branch of the logarithm) is self-adjoint on E, and as Xo(oo) = log J2

is diagonal, Xo(A) G g. Thus

e

tX°^x^ € G, and from the previous discussion, there exists a

factorization

etx°^=g+(t,\)g-(t,\) (2.110)

of type (2.103) satisfying (2.102). By uniqueness and self-adjointness we must have

0+(f,A) = 0 - ( U r (2.111)

and by reality, M

0

(A) = M0(A),

g±(t,\) = g±(tA) (2.112)

By the above calculations and by i?-matrix theory, we have proved most of the fol-

lowing basic result: