LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 33
provides the (unique) factorization of etX^x\
etxw = g+(t, A) (?_(t, A) , 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 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 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:
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