The Toda flow on Hessenberg elements of real, split simple Lie algebras.

In this work, we consider the Toda flow associated with compact/Borel decompositions of real, split simple Lie algebras. Using the primitive invariant polynomials of Chevalley, we show how to construct integrals in involution which are invariants of the maximal compact subgroup, and moreover, we show that the number of such integrals is given by a formula involving only Lie-theoretic data. We then introduce the space of Hessenberg elements, characterize the generic Hessenberg coadjoint orbits, and show that the dimension of such orbits is precisely twice the number of nontrivial invariants which appeared earlier. For the class of classical, real split simple Lie algebras, we construct angle-type variables which in particular shows that the Toda flow is Liouville integrable on generic Hessenberg coadjoint orbits. • We study the Toda flow on Hessenberg elements of real, split simple Lie algebras. • Nontrivial invariants of the maximal compact subgroup K in involution are constructed. • The formula for the number ν of such K-invariants involves only Lie-algebraic data. • Dimension of generic Hessenberg coadjoint orbits = 2 ν. • Toda is integrable on generic Hessenberg orbits for the classical simple Lie algebras. [ABSTRACT FROM AUTHOR]