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 g° 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) = ^ ,