Abstract
Let (G+R, ρ, V) be a regular irreducible prehomogeneous vector space defined over the real field R. We denote by P(x) its irreducible relatively invariant polynomial. Let V1 U V2 U . U V1 be the connected component decomposition of the set V-{x∈ V; P(x)=0}. It is conjectured by [MΓ4] that any relatively invariant hyperfunction on V is written as a linear combination of the hyperfunctions |P(x)|si, where |P(x)|si is the complex power of |P(x)|s supported on Vi. In this paper the author gives a proof of this conjecture when (G+R, ρ, V) is a real prehomogeneous vector space of commutative parabolic type. Our proof is based on microlocal analysis of invariant hyperfunctions on prehomogeneous vector spaces.
