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)) ,
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