Tyurin parameters of commuting pairs and infinite dimensional Grassmann manifold

Commuting pairs of ordinary differential operators are classified by a set of algebro-geometric data called “algebraic spectral data”. These data consist of an algebraic curve (“spectral curve”) Γ with a marked point γ∞, a holomorphic vector bundle E on Γ and some additional data related to the local structure of Γ and E in a neighborhood of γ∞. If the rank r of E is greater than 1, one can use the so called “Tyurin parameters” in place of E itself. The Tyurin parameters specify the pole structure of a basis of joint eigenfunctions of the commuting pair. These data can be translated to the language of an infinite dimensional Grassmann manifold. This leads to a dynamical system of the standard exponential flows on the Grassmann manifold, in which the role of Tyurin parameters and some other parameters is made clear.