LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 15

and hence

f^i-i

=0

Ad tx,Tv t r . y l

e e

= ^

+

r , ; r

+

. Y ] - [ 7 r _ r , 7 r _ X ]

= i([iir,.Y] + [r,i?A'])

(2.18)

= [Y,X]R.

Thus # is the Lie-algebra of G.

(b) Dual Lie-algebras and coadjoint orbits

Notice first that under (2.1) the loop M(A) = r^ becomes ^ ~ + ^o +

1 — A

Axz, where Ax = j2"^"\ A0 = ±±£- and A-X = A f = J 2 + M - I . One would hope

that elements of the form A-\z~x + Ao + A\z generated a coadjoint orbit consisting of

elements of the same form. As we will see, however, this is not the case and generically

elements of the form A-\z~x -f AQ -f A\z generate orbits consisting of elements of the form

——• + A-\z~x -f AQ + A\z with a polar singularity at z = 1 G S 1 .

Ay

z

Motivated by these considerations, we define

__ik

£*ng = {A : A(z) = — ^ + Ateg(z) = A(z) € gt(N, W) and

^reg(*) is a smooth loop on S 1 } .

In an obvious notation

^sing —pole — reg

(2.19)

(2.20)

We will show below that g*. is invariant under the coadjoint action of G, and the coadjoint

orbits of interest will be found to be subsets of g*

Elements A in g*. induce linear functionals on g_ through the non-degenerate pairing

(A,X)=j txA(z)X(z)^- (2.21)

Note that on g x g this pairing is ad-invariant,

(X,\Y,Z]) = -([Y,X),Z). (2.22)