LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 35
But if one writes (dg-(t, A) / dt) g~(t, A)
- 1
= 7_ -f 70 in the ^-notation (2.7), then
TT_(dg.{t, •) / dt) g-(t, - r 1 = 7 - + (70 - 7 - ( l ) )/2 (2.120)
On the other hand
*-,+(« , - ) -
1
( ^ i ) = constant = \g+{t, , = !)"» ^ + *
= L _ ( t , z = I ) "
1
±g_(t, z = 1) , by (2.102),
& at 12121")
1 / 7 """
^
= 2^Ttg~^'z = ^ ^g-^'z = 1 ^~ 1 ' b y d i a n a l i t y
= - (
7
_ ( l ) - h
7
o ) .
Adding (2.120) and (2.121) we obtain (2.119).
To prove (2.118) note first from (2.114) that (1
\2)M(t,
A) has an analytic contin-
uation to Re A 0 and to Re A 0, and grows at most quadratically as A —• oo. By
Liouville,
~ Bo + XB1 +
X2B2
M(t,X) = j ^
for suitable matrices Bo, B\ and B2. Letting A —• 00 in (2.114), and using the diagonality
of g+(t, A = 00), we see that B2 =
—J2.
Letting A —• 0, we find
Bo = M(*,0) = y+^O)"
1
/(/+(*, 0) = I .
From (2.111), for-A ifl,
M ( t , A r = y
+
( f
t
A r (
f + A
^
A 2 J
^
T _ \ M _ \
2 J2
= g.(t,X)C ^ _ °
A 2
){g-(t,\)Y (Mo is skew)
= M{t, X) ,
and hence Bi = -B{. But using (2.112), we find M{t,X) = M(t,X), and so Bi is real. It
follows that
~ ^
I-XM(t)-X2J2
M(t,X) = ^ ,
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