Toda lattices with indefinite metric II: Topology of the iso-spectral manifolds

We consider the iso-spectral real manifolds of tridiagonal Hessenberg matrices with real eigenvalues. The manifolds are described by the iso-spectral flows of indefinite Toda lattice equations introduced by the authors [Physica, 91D (1996), 321-339]. These Toda lattices consist of $2^{N-1}$ different systems with hamiltonians $H = (1/2) \sum_{k=1}^{N} y_k^2 + \sum_{k=1}^{N-1} s_ks_{k+1} \exp(x_k-x_{k+1})$, where $s_i=\pm 1$. We compactify the manifolds by adding infinities according to the Toda flows which blow up in finite time except the case with all $s_is_{i+1}=1$. The resulting manifolds are shown to be nonorientable for $N>2$, and the symmetric group is the semi-direct product of $(\ZZ_2)^{N-1}$ and the permutation group $S_N$. These properties identify themselves with ``small covers'' introduced by Davis and Januszkiewicz [Duke Mathematical Journal, 62 (1991), 417-451]. As a corollary of our construction, we give a formula on the total numbers of zeroes for a system of exponential polynomials generated as Hankel determinant.