Compactified isospectral sets of complex tridiagonal Hessenberg matrices

Abstract

The completed flows of the complex Toda lattice hierarchy are used to compactify an arbitrary isospectral set $J$^ of complex tridiagonal Hessenberg matrices. When the eigenvalues are distinct, this compactification, as found by other authors, is a toric variety; it is the closure of a generic orbit of a complex maximal torus inside a flag manifold. This torus becomes a direct product, A, of a nonmaximal diagonal subgroup and a unipotent group when eigenvalues coincide. We describe the compactification of $J$^ in this case as the closure of a generic orbit of A. We are interested mainly in the structure of its boundary, which is a union of nonmaximal orbits of A. There is a one-to-one correspondence between the connected components of the intersections of $J$^ with the lower-dimensional symplectic leaves and the faces of the moment polytope where at least one vertex is a minimal coset representative of a certain quotient of the Weyl group.