The Orders of Invariant Eigendistributions

Abstract

In a series of papers, Harish-Chandra studied the invariant eigendistributions and in particular established a fundamental theorem which states that any invariant eigendistribution on a connected semisimple Lie group is locally L (cf. [H]). It may be available to reconsider this result from the view point of microlocal analysis. On the other hand, recently Hotta and Kashiwara [HK] have shown that the system of differential equations which governs an invariant eigendistribution on a semisimple Lie algebra is regular holonomic (=a holonomic system with regular singularities in the sense of [KK]). Among other things they showed, by using this result, that the holonomic system in question corresponds to the intersection cohomology complex defining Springer's representations of the Weyl group through the Riemann-Hilbert correspondence. In this paper we examine a microlocal property of the invariant eigendistributions. The results of this paper is quite unsatisfactory in comparison with those mentioned above. But the author hopes that our attempts will be developed in future. We now explain the contents shortly. In the first half we consider the homonomic system M^ which governs an invariant eigendistribution on a connected linear semisimple Lie group. An invariant of a holonomic system is the set ordA(u) of the orders along each irreducible component A of the characteristic variety of the system in question. Here u is a section of the system on the generic points of A. We attempt to determine ord^(w) for the system MI. Unfortunately, we cannot do it for every irreducible component A of the characteristic Ch(*SKx) but if an irreducible component A of Ch(^) satisfies the condition (A) in (3.1), we can calculate the orders along A. In this case, though c5^z is not a simple holonomic system in the sense of Sato-Kashiwara, ord^w) along such an irreducible component A consists of only one element 0. This is the main result of the first half (Theorem (3.4)). It rarely occurs that