On the Geometry of Differential Equations Associated with the Theta-zerovalue

Abstract

with t being a symmetric complex nXn matrix and the imaginary part of t being positive definite, when n = 2. Their approach is to analyse the structure of linear differntial equations of infinite order which the theta-zerovalue satisfies. This approach was first advocated by Sato ([7]) and later pursued by Sato, Kashiwara, Kawai and Takei ([4], [6], [8], [9], etc.). In this paper we present some geometric results which are needed to extend the results of [4] for an arbitrary n. Although we have not yet obtained the complete generalization in this paper, we plan to discuss the more analytic aspect of the problem, using the algebraic and geometric results shown here. The plan of this paper is as follows; in §2, we first introduce (n + 1) X (n + 1) matrices PI, . . . , Pn, Q i , . . . , Q« of linear differntial operators of finite order such that the following relations hold: