A Class of Morse Functions on Flag Manifolds

Given a positive definite symmetric matrix in one of the groups SL(n, R) or Sp(n, R), we analyse its actions on Flag Manifolds, proving these are diffeomorphisms which admit as strict Lyapunov functions a special class of quadratic functions, all of them Z2 -perfect Morse functions. Stratifications on the Flag Manifolds are provided which are invariant under both the diffeomorphisms and the gradient flows of the Lyapunov functions.