Blow-ups of the Toda lattices and their intersections with the Bruhat cells

Abstract

We study the topology of the set of singular points (blow-ups) in the solution of the nonperiodic Toda lattice defined on real split semisimple Lie algebra $\mathfrak g$. The set of blow-ups is called the Painlev\'e divisor. The isospectral manifold of the Toda lattice is compactified through the companion embedding which maps themanifold to the flag manifold associated with the underlying Lie algebra $\mathfrak g$. The Painlev\'e divisor is then given by the intersections of the compactified manifold with the Bruhat cells in the flag manifold. In this paper, we give explicit description of the topology of the Painlev\'e divisor for the cases of all the rank two Lie algebra, $A_2,B_2, C_2, G_2$, and $A_3$ type. The results are obtained by using the Mumford system and the limit matrices introduced originally for the periodic Toda lattice. We also give a Lie theoretic description of the Painlev\'e divisor of codimension one case, and propose several conjecturesfor the general case.