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 ( F q ) related to the maximal compact subgroup K of the group G ˇ with g ˇ = Lie ( G ˇ ) over the finite field F q . Here g ˇ 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.