The Schur–Sato Theory for Quasi-elliptic Rings

The notion of quasi-elliptic rings appeared as a result of an attempt to classify a wide class of commutative rings of operators arising in the theory of integrable systems, such as rings of commuting differential, difference, and differential–difference operators. They are contained in a certain noncommutative “universal” ring, a purely algebraic analog of the ring of pseudo-differential operators on a manifold, and admit (under certain mild restrictions) a convenient algebraic–geometric description. An important algebraic part of this description is the Schur–Sato theory, a generalization of the well-known theory for ordinary differential operators. Some parts of this theory have been developed earlier in a series of papers, mostly for dimension $2$. In this paper we present this theory in arbitrary dimension. We apply this theory to prove two classification theorems of quasi-elliptic rings in terms of certain pairs of subspaces (Schur pairs). They are necessary for the algebraic–geometric description of quasi-elliptic rings mentioned above. The theory is effective and has several other applications, including a new proof of the Abhyankar inversion formula.