Opérateurs différentiels de Shimura et espaces préhomogènes. (Shimura differential operators and prehomogeneous spaces) (Q1099208)

The purpose of this paper is to reformulate \textit{G. Shimura}'s result [Invent. Math. 77, 463-488 (1984; Zbl 0558.10023)] from a new point of view, to prove an extended version of Shimura's theorem in a unified manner and to give an application for local zeta distributions on prehomogeneous vector spaces. G. Shimura found a remarkable formula in that paper. In proving the formula, he utilized the property that the action of the parabolic subgroup of a reductive group is prehomogeneous though the term ``prehomogeneous'' was not used. The proof was carried out in a case-by-case study. The authors of this paper succeed in arriving at Shimura's results by using the properties of prehomogeneous vector spaces of commutative parabolic type, which had been studied in their previous paper [Math. Ann. 274, 95-123 (1986; Zbl 0568.17007)]. Moreover, they prove a functional equation of vector-valued zeta-distributions associated to the prehomogeneous vector spaces of commutative parabolic type.