LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 29

where Er(-) is the elementary symmetric function of degree r in the entries of M, Er(tM) =

trEr(M)

for all t G R. Thus

Er(-z-lA-i

+ (AP+A-1 - A

0

) + ^(Ao - A i ) + *

2

Ai) = ] T Jr*2* . (2.88)

k=-r

(Note that the notation is consistent: J, ,

= 1

Ji„|

= 1

IrkZkrjn~r

n-?u p ^ r = /rib-) It follows

that the number of nontrivial integrals is given by

N

] T ( 3 r + 1) = 3iV(iV + l)/ 2 + AT

r = l

Simple computations show that the coadjoint invariants {tr

AfL11

tr(A

p

—

Ao)AIL1}ik,k,N a r e

equivalent to {Ir,-r, Ir',-r'+i}ir,r'N•

a n

d the coadjoint invariants

{tr Af, tr AoAf }ijt,fc'N are equivalent to {/r,2r, -f"r',2r'-i}ir,r'N- We thus obtain

(3N(N + l)/ 2 + iV) - 4iV = 3N(N - l)/ 2

nontrivial integrals,

{^r*}lri V , - r + 2 f c 2 r - 2 ( 2 . 8 9 )

on the orbit OA- We construct the remaining

(

2~ commuting integrals from the entries

of Ap. By (2.86) it is enough to construct N(N — l)/ 2 combinations of the entries of Ap

that commute amongst themselves.

Note first that (Ap)ij can be rewritten in the form

,z*-l.

(A,)ij = *y(A ) =j ( ^ 3 ^ ) t r ( A (

2

) E

J

, ) ^

2

where £^;i is the elementary matrix tjtj. By differentiation

(2.90)

dQij(A) = {£-r^)Ejieg, (2.91)

and we find

Rd$ij{A) =

(z-z-1)2Eji

. (2.92)