Singular structure of Toda lattices and cohomology of certain compact Lie groups

We study the singularities (blow-ups) of the Toda lattice associated with a real split semisimple Lie algebra g. It turns out that the total number of blow-up points along trajectories of the Toda lattice is given by the number of points of a Chevalley group K(Fq) related to the maximal compact subgroup K of the group Gˇ with gˇ=Lie(Gˇ) over the finite field Fq. Here gˇ is the Langlands dual of g. The blow-ups of the Toda lattice are given by the zero set of the τ-functions. For example, the blow-ups of the Toda lattice of A-type are determined by the zeros of the Schur polynomials associated with rectangular Young diagrams. Those Schur polynomials are the τ-functions for the nilpotent Toda lattices. Then we conjecture that the number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable. We also discuss the case of periodic Toda lattice in connection with the real cohomology of the flag manifold associated to an affine Kac-Moody algebra.