Whittaker modules and hyperbolic Toda lattices

Let $\text{sg}$ be a complex finite-dimensional simple Lie algebra and let ${\text{sg}}_l$ be the corresponding generalized Takiff algebra. This paper studies the affine variety $\text{ssf}+{\text{sb}}_l$ where $\text{ssf}$ is similar to a principal nilpotent element of $\text{sg}$ and ${\text{sb}}_l$ is a subalgebra corresponding to the Borel subalgebra $\text{sb}$ of $\text{sg}$. Inspired by Kostant’s work then we deal with two questions. One of them is to construct the Whittaker model for the $G_l$-invariants of symmetric algebra $S({\text{sg}}_l)$ where $G_l$ is the adjoint group of ${\text{sg}}_l$ and $G_l$ acts on $S({\text{sg}}_l)$ by coadjoint action, and then to classify all nonsingular Whittaker modules over ${\text{sg}}_l$. Another one is to describe the symplectic structure of the manifold $Z\subseteq \text{ssf}+{\text{sb}}_l$ of normalized Jacobi elements. Then the Hamiltonian corresponding to a fundamental invariant provides a class of hyperbolic Toda lattices. In particular, a simplest example describes the state of a dynamical system consisting of a positive mass particle and a negative mass particle.