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Let $\sg$ be a complex finite-dimensional simple Lie algebra and let $\sg_l$be the corresponding generalized Takiff algebra. This paper studies the affinevariety $\ssf+\sb_l$ where $\ssf$ is similar to a principal nilpotent elementof $\sg$ and $\sb_l$ is a subalgebra corresponding to the Borel subalgebra$\sb$ of $\sg$. Inspired by Kostant's work then we deal with two questions. Oneof them is to construct the Whittaker model for the $G_l$-invariants ofsymmetric algebra $S(\sg_l)$ where $G_l$ is the adjoint group of $\sg_l$ and$G_l$ acts on $S(\sg_l)$ by coadjoint action, and then to classify allnonsingular Whittaker modules over $\sg_l$. Another one is to describe thesymplectic structure of the manifold $Z\subseteq\ssf+\sb_l$ of normalizedJacobi elements. Then the Hamiltonian corresponding to a fundamental invariantprovides a class of hyperbolic Toda lattices. In particular, a simplest exampledescribes the state of a dynamical system consisting of a positive massparticle and a negative mass particle.

Whittaker modules and hyperbolic Toda lattices