LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 23

For X = YL°}L-oo XjZ-* = 7r+X -f TT-X G £, a simple computation shows that

$(etn+x etir~x)

lt=o

(-(*

+

X)(0) , A',) , ( ( i . j r ) ( o o ) , - X - , ) ) • (2.59)

*(*,

- ±

Also it is easy to see that j is onto g (for this purpose it is enough to consider X of the

form - ^ 4- • • • + X2Z2). Thus j* is injective from g* into £*.

To compute f* we identify £* with g through the standard nondegenerate pairing,

(((A/,JO,(i?,S)), ((M'J{')AR\Sf)) = tr M M ' + tr Ji Aw 4- tr JLR'+ tr 5 5 ' . (2.60)

Then for a = ((M,/v), (#,S) ) G £j ~ £Q,

« I ) = ( a ^ ( x ) )

= tr KXX - tr M T T + X ( 0 ) - tr SX.x + tr i?7r_X(oo) .

On the other hand, for 0 = Bp/(z — 1) -f B - i z - 1 + #0 + #12, a now familiar computation

shows that

(/?,X) = tr(£

0

- Bp)w+X{0) + tr £-1X1 + tr 507r_(oo) + tr BXX-X .

We conclude that

f*((M,K) , (fl,5) ) = ^ ^ + - + R-zS , (2.61)

z — \ z

or alternatively,

* * ( ( ( £ , - B o ) , £ - i ) , ( B

0

, - S i ) ) = ^ + ^ - + J 3

0

+ B i * . (2.62)

Thus the orbits through points f*a G g* are precisely the orbits considered in (2.41). To

compute the dimension of the coadjoint orbit through t*a, it is sufficient, by Lemma 2.43,

to compute the dimension of the orbit through Q in g*. A standard computation shows

that for ((g,u),(h,v)) G Go

Ad^,

u )

,

( M ) )

((M,A-),(tf,S) )

(2.63)

= ((gMg-1 + [u,gKg-1],gKg-1),(hRh-1 +[v,hSh-1}, hSh-1)) ,