20 P. DEIFT, L. C. LI, AND C. TOMEI

for all a € £*. But for a,/3 € £*,

d ( F o ^ ) ( a ) ( / 9 ) = | Fo(f*(a + tf3)

dF(f(a))W))

= 4{dF(r(a)))(/3).

Thus

d(For)(a) = j(dF(r(a)))€l1 ,

and hence F o f* is smooth on g* We now have

a([d(F o *•)(«) , d(G o ^*)(a)]) = a{[/(dF(4m{a))), t(dG(f*(a))})

= a ( # [ W ( a ) ) , W ( a ) ) ] ) )

= * ' ( a ) ( [ W ( a ) ) , W ( a ) ) ] ) ,

as desired. D

The point is this. From (iv) we have 0^*(a) C t*{Oa) for all a G #*, and if $ is onto

C?2, then

O ^ ( « ) = 0 * ( O a ) • (2.45)

Moreover, if f is onto g , then £* is injective and hence bijective from 0£*(o) onto O

a

;

in particular they have the same dimension. We will construct a homeomorphism $ from

G\ = G onto a finite dimensional Lie-group G*2, with ^ surjective, for which the finite

dimensional orbits 0^*(a) given by (2.45) are precisely the orbits OA in (2.41). In this way

the computation of the orbits is reduced to an equivalent problem for a finite dimensional

group.

Let Go denote the identity component of the direct sum of two copies of the semi-direct

product G£{N,R)ad x g£(N,R),

Go = {((g,u),(h,v)) :g,he G£(N,R), det g 0, det ft 0 , u, v e g£(N,R)} , (2.46)

((?, i/), (ft, v)) o ((?', u1), (/*', v')) = ((g'g, u' + ^ ( ( / T 1 ) , (h'h, v' + ft'i^ft')"1)) • (2-47)