Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities

Abstract

Let W be a finite Coxeter group acting linearly on $R^n$. In this article we study support properties of W-invariant partial differential operator D on $\R^n$ with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(supp Df) = conv(supp f) holds for all compactly supported smooth functions f so that conv(supp f) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.