Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

Let X = G/H be a semisimple symmetric space of non-compact style. Our purpose is to construct a compact real analytic manifold in which the semisimple symmetric space X = G/H is realized as an open subset and that $G$ acts analytically on it. By the Cartan decomposition G = KAH, we must compacify the vectorial part A.$In [6], by using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space G/H is realized as an open subset and that G acts analytically on it. Our construction is a motivation of the Oshima's construction and it is similar to those in N. Shimeno , J. Sekiguchi for semismple symmetric spaces. In this note, first we will inllustrate the construction via the case of SL (n, R)/SO_e (1, n-1) and then show that the system of invariant differential operators on X = G/H extends analytically on the corresponding compactification.