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 + C° + 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)