LOOP GROUPS, INTEGRABLE SYSTEMS, AND RANK 2 EXTENSIONS 9
(3) as anticipated from [MV], the flow t H- M(t,\) = {I - XM(t) - A2 J 2 ) / ( l - A2)
induces an integrable Hamiltonian flow t »-* M(t) on the dual Lie-algebra o(N)* equipped
with the standard Lie-Poisson structure.
In Section 3 we consider the general case where (1.24) may fail and 5 = 5+ U 5_
is any decomposition satisfying (1.19). As we show, this leads to a loop algebra, and a
Riemann-Hilbert factorization problem, on a skeleton S = Uj=-7 ^ i composed of lines S j
(asymptotically) parallel to iR with So = iR and Ej = —E_j. All the results of Section 2
extend to this situation, but we focus on the novel aspects of the Riemann-Hilbert problem.
The authors in [MV] also consider the more general problem S = Ejt tr XicJX'[+v
where Xk now lies in the Stiefel manifold Vm%iw of in x N matrices satisfying XkX^ = Im.
This leads to a factorization problem for quadratic matrix polynomials of the form
A2 J 2 + AM, - XLiXk-i . (1.42)
In Section 4, we concentrate on the case where m = 1, leaving the more general problem,
with its many connections to the work of Adams, Hamad and Previato [AHP] and Schilling
[S], to a later publication. We focus in particular on the billiard ball problem in an elliptic
region E = {x :
(x,C~2x)
1} in J R ^ , where C is positive and diagonal. Although the
variational functional S = Ylk \Xk "~ ^fc-il* (xk*C~2Xk) = 1, is not of the above type,
Moser and Veselov show that the solution of the billiard ball problem in E also reduces to
a polynomial matrix factorization problem for a quadratic polynomial
1(A) = y ® y + A a r A y - A2C2 , (1.43)
where
(x,C" 2 x ) = l and (y,y) = l . (1.44)
Our results in Section 4 are the following.
(1) As before, the coadjoint orbit of G through A{\) = L(X)/(1 - A2) provides a sym-
plectic structure for the problem and the analog of the factorization (1.3S) again leads to
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