18 P. DEIFT, L. C. LI, AND C. TOMEI
Now take A = j ^ -f -p^- -\ h Aezi. From the analyticity properties of g±, it follows
that
and
r '
( ^ A ( / - ) =
r
f
T
+ ^ C
J
V (2.38)
oo
for suitable matrix coefficients Bp, Bj, C
p
, C'j. Hence
- l
((^i^+)reg)_=
Y
B^
and
£
({gZlAg-U%)+
= YJC^
The result now follows from (2.29) and (2.30). D
In the remainder of this subsection we consider only the case k = £ = 1,
A=-^-
+ ^ +
A0
+ A1z.
z 1 z
The above formulae reduce to
(Ad,A)(z) = Br-Bo + + B..Z-1 + C0 + C,z ,
y z 1
and a straightforward computation of the constants yields, finally, the formula
(AdgA)(z) = ^ { [ M 0 ) - 1 s ; ( 0 ) , M 0 r 1 . 4 - i
f f +
( 0 ) ] +5+(0)~ 1 (A
p
- Ao)p+(0)
- [
5
_(oo)- 1
S
'_(^),
?
_(^)- 1 .4
l f f
_(oo) ] +
s
_(rc)- 1 A
o ?
-(0) }
| g +
( 0 ) - M _
l g +
( 0 )
z
- [g-(oo)-1 g'_(oc),g-{oo)-1 A^j-{oo)] + g-{oo)~x A0g-{oo)
+ g.{oo)-1Alg-{oo)z, (2.39)
where
4(0) = ^ ( z = 0) and ff'-(oo) = ^ - ( z = oo) . (2.40)
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