We construct action-angle variables for the Toda flow on generic orbits of the coadjoint action of The Toda flow (see [14]) on (WZn, w = Xy=l dx' A dy') is generated by the the lower triangular group on its dual Lie algebra.Hamiltonian Flaschka [7] (see also Moser [ll]) showed that the flow is completely integrable by making the change of variables , n , THE TODA FLOW 185 follows directly from the invariance properties of these quantities with respect to the above subgroups. The commutativity of the wrk, however, as well as the commutation relations between the A,, and the wr,kt does not follow directly from invariance properties, and relies on certain ad hoe calculations whose Lie theoretic significance we have not yet fully understood. In Section 4, we solve an inverse problem, which we then use in Section 5 to prove that the map from a neighborhood of an invariant set to the action-angle variables is a symplectic diffeomorphism. In Section 5, we also show how to solve the equations generated by the Hamiltonians A r k using the QL factorization for matrices, in the manner of Symes [13]. In Section 6, we show how to extend the ideas of the previous sections to nongeneric orbits, in particular, orbits through pentadiagonal matrices. In Section 7, from a different point of view, we show that the Toda flow on a, dense set of orbits can be mapped injectively into a product of independent Toda flows on tridiagonal orbits. Since the Toda flow on tridiagonal orbits is completely integrable by the classical results of Flaschka, this gives another interpretation of integrability.In [12], Symes introduced and discussed the appropriate generalization of the Toda flow to general, split, semi-simple Lie algebras. The reader will see that the methods of this paper extend naturally to prove complete integrability of the Toda flow for the classical simple Lie algebras, so(n,R) and sp(n,U), but we provide no details.An announcement of some of the results of this paper appeared in [5]. For the reader's convenience, we include a list of symbols and definitions at the end of the paper.
Variables, Groups and OrbitsAll the computations that follow are for real n X n man-ices. For an n X n matrix M , let = {Mi,: k + 1 5 i 5 n, 1 s j 5 n -p } be the ( nk) X ( np ) matrix obtamed by deleting the first k rows and the last p columns of
Erk( M)A"-2k-r. r = O
As we shall see, the ratios E r k ( M ) / E O k ( M )provide the integrals for the Toda flow. They are, of course, just the symmetric functions of the A,, described in the introduction.Note that Em( M ) = ( -1)" and E,.,~( M ) = det( M ) " L k for n even, 1 g k g $ n 7 k + l = (-1) det(M),-k for n odd, 1 =< k =< $(n -1).
'By the expressions ( M -A ) and ( M -A ) , we, of course, mean ( M -X I ) and ( M -A I ) h .respectively. where I is the n X n identity matrix. 186 P. DEIFT, L. C. LI, T. NANDA, AND C. TOME1 In particular, Eok( M ) depends only on entries in the last k mws and the first k columns of M . All these entries lie strictly below the main diagonal. The polynomials Pk(M, A) ...
