LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 17
and
TlA =
iji^k +
{Aregh
+ (A
r e g
)
0
- i(Ap)diag*i . (2.30)

R e m a r k . The use of the symbols 7r±A above involves a slight but irrelevant abuse of
notation as A(l) need not be diagonal, and so A may not lie in g where TT± are defined.
For convenience in this subsection we replace g- by gZ in the basic factorization for
g £ G. Thus
9^9+gZ1
(2.31)
and (2.17) takes the more symmetrical form
Ad9A" = g+(n+X)gI1 + g-(n_X)gZ1 . (2.32)
In all the other parts of the paper we retain the factorization as in (2.14), which is more
convenient for the analytical questions that arise.
For A G g*. , and g 6 G, g±lAg± also lie in g*. , and from (2.32)
AdgA = 7rl(g+1Ag+) + T*_(gZ1Ag.) (2.33)
Lemm a 2.34. Ad maps g*. into itself. Moreover for any k,£ £ Z + , the set
{A : A = - ^ - + ^ + + Atze\ is Ad*-invariant. (2.35)
z 1 z
Proof: To prove Ad„7* C a*. it is enough by Lemma 2.28 to show that
^ 92-smg ising to J
( ( f f ; 1 ^ + )
P
)
d i a g
- ((gZ'Ag.),)^ = 0 (2.36)
But
((g^Agi),)^ = (7i 1 (l).4
p
(l))
dia g
= (A
p
)
d i a g
,
as g±(l) are diagonal, which proves (2.36).
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