Convex polytopes and duality in the geometry of the full Kostant-Toda lattice.

Abstract

Our study describes the structure of the completely integrable system known as the full Kostant-Toda lattice in terms of the rich geometry of complex generalized flag manifolds and the information encoded in their momentum polytopes. The space in which the system evolves is a Poisson manifold which is essentially the dual of a Borel subalgebra of a Lie algebra, and the symplectic leaves are the coadjoint orbits. We extend the results of Ercolani, Flaschka, and Singer in (4) in which an embedding of an isospectral submanifold of the phase space into the flag manifold is used to study the geometry of the "generic" compactified level sets of a particular family of constants of motion. In a detailed analysis of the full Sl(4,C) Kostant-Toda lattice, we consider all types of level sets, in particular those which do not satisfy the genericity conditions of (4). The breakdown of these conditions is reflected in the types of nongeneric strata to which the torus orbits in a "special" level set belong. This degeneration corresponds to certain decompositions of the momentum polytopes, which we explain in terms of representation theory. We discover a fundamental two-fold symmetry intrinsic to this geometry which appears in the phase space as an involution preserving the constants of motion, and we express it in terms of duality in the flag manifold and the pairing between a representation of a Lie algebra and its dual. Several chapters are devoted to the study of a double fibration of a generic symplectic leaf by the level sets of two distinct involutive families of integrals for the full Sl(4,C) Kostant-Toda lattice. We describe the symmetries of these two fibrations and determine the monodromy around their singular fibers. Finally, we show how the configuration of the lower-dimensional symplectic leaves of the Poisson structure in this example is revealed in the geometry of the flag manifold and its momentum polytope.