Toda lattice, cohomology of compact Lie groups and finite Chevalley groups

In this paper, we describe a connection that exists among (a) the number of singular points along the trajectory of Toda flow, (b) the cohomology of a compact subgroup $K$, and © the number of points of a Chevalley group $K({\mathbb{F}}_q)$ related to $K$ over a finite field ${\mathbb{F}}_q$. The Toda lattice is defined for a real split semisimple Lie algebra $\mathfrak{g}$, and $K$ is a maximal compact Lie subgroup of $G$ associated to $\mathfrak{g}$. Relations are also obtained between the singularities of the Toda flow and the integral cohomology of the real flag manifold $G\mathrm{/}B$ with $B$ the Borel subgroup of $G$ (here we have $G\mathrm{/}B=K\mathrm{/}T$ with a finite group $T$). We also compute the maximal number of singularities of the Toda flow for any real split semisimple algebra, and find that this number gives the multiplicity of the singularity at the intersection of the varieties defined by the zero set of Schur polynomials.