ADVANCES IN MATHEMATICS 34,195338 (1979) The Solution to a Generalized Toda Lattice and Representation Theory BERTRAM KOSTANT* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 CONTENTS 0. Introduction 0.1. Statement of the main results 0.2. A brief description of Sections l-7 0.3. General comments 1. Poisson commutatierity of translated invariants 1.1. A formula for the Poisson bracket 1.2. The case of a subalgebra of a semi-simple Lie algebra 1.3. Relating cent 8’ to S(g)’ 1.4. The theorem [I’, Jf3 = 0 for Lie summands 1.5. The case of a Bore1 subalgebra where f = C e-,. 1.6. The commutativity of certain vector fields on a;’ 2. The variety Z of normalized Jacobi elements 2.1. The variety 8, = f + d and invariants I E So 2.2. The vector fields II, I E So, on Z 2.3. The subvarieties Z(y) Z; Z and IV-conjugacy 2.4. The Bruhat-Gelfand decomposition, the isomorphism j3,: Q* -+ Z(r), and the rationality of Z(y) in the complex case 2.5. The foliation Z = U Z(r) in the complex case 2.6. The isomorphism b(W): G1”,, + Z(y) in the complex case 3. The parametrization Z s H x A, in the real case 3.1. The polar decomposition and centralizers # 3.2. The relation Gv n RHN = G,u for y E Z and the isomorphism Z(y) z [w’ 3.3. The open Weyl chamber R, and Z 3.4. The normalizer G of 9 in Ad gc and the subset G(*) = s(K)G* C G 3.5. The element ii,(w) and the key relation & n G(*) = Gow 3.6. The isomorphism &,): G,” + Z(y) 3.7. The isomorphism H x A, --f Z, (g., zoo) t+ n(g)w * Supported in part by the NSF under Grant MC% 7804007. 195 OOOl-8708/79/120195-144$05.00/O Copyright 0 1979by AcademicPress. Inc. AU rights of reproductionin any form reserved. 196 BERTRAM KOSTANT 4. The isospectral leaf Z(y) as a complete flat afinely connected manifold 4.1. The union Z = U Z(y) as a foliation 4.2. The flat a@e connection in Z(y) 4.3. The &aps &) and &, as isomorphisms of complete affinely connected manifolds 5. Representations and the functions @A(gO , w,; t) 5.1. The module V”, h ED, as a real Hilbert space 5.2. The normalization of the lowest weight vector ‘&A = S,(K)& 5.3. The equation ii(a) h(a) n(a) = s~(K)-+~-,(w) a,irt(w)-l 5.4. The .polar decomposition of h(g)‘lz n(g) ii,(w) 5.5. Formulas for d(w) = g;’ p(g., w,) in terms of ii&), as a limit, and its independence of g, . Also formula for g,,A in terms of n(g) and sum formula for h(g exp twr 5.6. Relation to Moser’s treatment of the Toda lattice in [19] 5.7. Sums over Y and the completion of U(m) and U(G) 5.8. The polynomials p(s, w,) and formulas for S,(W), A,(W)-‘, and B-~(W) 5.9. The numbers C,J and formulas for d(zur and d(w)-~h 5.10. The definition of @A(g, , w,,; t) and the result h(g exp(-t)w)A = @A(g. , w,; t) 5.11. The limiting behavior of log h(g exp( - t)w)” as t -+ + cc 6. The symplectic structure of (Z, oz) and the integration of !k, I P S(g)” 6.1. The symplectic structure computed in terms of Poisson brackets 6.2. The symplectic structure on coadjoint orbits 6.3. The coadjoint theory transferred to a* 6.4. The symplectic manifold (Z, wz) as a translated coadjoint orbit 6.5. Formula for II 1 Z in terms of canonical coordinates 6.6. Complete real polarizations F and integration of &, i E CFW 6.7. The formula fl(,)(g exp(-t)z) = exp tfI . y and the formula yi(exp tb . y) = log h(g exp( - t)z)-“i 6.8. The special case where I = I1 = Q/2 7. D&ouement; the formula for qi(x(t)) 7.1. Hamilton’s equations, the Hamiltonian H = xppia/2m, + r,& + ... + r&s, and the bilinear form BH 7.2. Reducing the integration of 6~ to the integration of Qs 7.3. The fixing of Q 7.4. Examples of where the & define a Dynkin diagram and the tie-up with (Z, oz) 7.5. The main result; the formula for q((x(t)) 7.6. The scattering of the generalized Toda lattice in terms of g, and d(w) 7.7. Application to the standard Toda lattice and the recovery of Moser’s result on its scattering, limr+*m q&(t)) 7.8. The three-body problem &., pie/2mi + eal+z + e’W4s + e*% worked out 0. INTRODUCTION 0.1. One of the main results of this paper concerns the existence of certain classical mechanical systems, generaliking the (finite, nonperiodic) Toda lattice (see, e.g., [8, 10, 19, 24, 2.51) w h ic h one can explicitly integrate for all values of time. These systems are related to Dyqkin diagrams. One TODA LATTICE AND REPRESENTATION THEORY 197 knows these diagrams classify semi-simple Lie groups. The integration of the system associated to a Dynkin diagram is expressible in terms of the finite- dimensional representations of the corresponding group. In fact, more than that is true. The integration of the system and the theory of the finite-dimen- sional representation theory of semi-simple Lie groups are in a sense equivalent. Indeed, the integration of the system completely determines and is determined by the weight structure of the fundamental representations of the corresponding group. In more detail, recall the Hamilton-Jacobi theory as it would apply to a system of particles moving on a line with respect to some potential. We envision then n particles, say with masses mi , i = l,..., n, and lR2n with linear canonical coordinate functions pi , qi , i, j = l,..., 71(i.e., phase space), so that any x E lR2n is a (classical) state of the system andp,(x), qi( x ) are, respectively, the momentum and position of the ith particle in the state X. Assume the potential U is smooth ( i.e., U E C@(lR2~)), and of course, depends only on the q’s (i.e., (a/Zppi)U = 0, i = 1, 2,..., n). The Hamiltonian HE Coc(lR2”) of the system is then given by (0.1.1) H = g1 g 1 + 0’. Now if x E lR2n and x(t) E R2n is the state of the system at time t such that x(0) = &then z(t) is determined by Hamilton’s equations. That is, - z$ (#) = +jl;f@)) , p&(t)) = mj dq$(t)) . (0.1.2) The fundamental problem of the Hamilton- Jacobi theory is to integrate (0.1.2), that is, to determine the function t + x(t). It is clear from the second equation in (0.1.2) that it is enough to determine the position functions t ---f qi(x(t)), j = l,..., n. For a general potential function U of course one cannot do this. It is one of the main results of this paper to show that for certain very special U having to do with Dynkin diagrams one can explicitly integrate (0.1.2) using representation theory. Let 9 C Cm(lR2~) be the n-dimensional vector space spanned by qi , j = I ,-.., n.LetZ<nandlet&~Z?,i= I,..., 1, be I linear independent functions. Thus we may write i = I,..., I, (0.1.3) where (uu) is a constant 1 x n matrix. The potentials we shall consider are those of the form U = ideal + ... + rLedz, (0.1.4) 198 BERTRAM KOSTANT where the ri are some positive constants. For example, if I = n - 1, m, = ri = 1 and & = qi - qi+r , i = I,..., I, then the system is the Toda lattice. The interaction between the particles is that of exponential forces between nearest neighbors. (Throughout this paper “Toda lattice” will mean “finite nonperiodic Toda lattice”.) What is crucial for us in our result here is the geometry of the JI’s, that is, the lengths of the vectors &E 2 and the angles between them. We hasten to add that there is in fact a natural geometry in 2, defined by the Hamiltonian H. Indeed let [q~,#] E Cm(lRz~) be the Poisson bracket for any pair of functions ‘p, # E C”(Ra”). It is easy to see that if v, # E 9 then [q~, [I/J, H]] is a scalar times the identity function 1. A positive definite symmetric bilinear form BH is then defined in .J?Jby the relation WP), W = Pa [vi WI. (0.1.5) One has in fact B&i , qj) = %h . (0.1.6) We will say that the #‘s define a Dynkin diagram if there exists a real split semi-simple Lie algebra 9 of rank 1 with an invariant bilinear form Q, and with a split Cartan subalgebra k with simple roots 011,..., iyr such that Wh 9 ~4) = Q(ai 34 (0.1.7) for i,j = l,..., 1. In such a case, as one knows, the possible angles between the #‘s are 90, 120, 135, or 150”. The Toda lattice satisfies this condition. Here 9 is the Lie algebra of SZ(n, R). The angle between qiel - qi and qi - qi+l is 120”.