Microlocal Calculus and Fourier Transforms

§ 1. In this note we shall explain the general scheme for calculation of the Fourier transforms of relatively invariant hyperfunctions on regular prehomogeneous vector spaces by Microlocal Calculus. The details of the theory will appear in [1], [3] and [6]. The principal purpose of this note is to state Theorem 6. A prehomogeneous vector space over the complex number field C is by definition a triple (G, p, V) of an affine algebraic group G, a finite dimensional vector space V and a linear representation p of G on V, all defined over C, such that there exists a point xEiV, (which is called a generic point of V), whose G-orbit p(G) -.ris dense in V. A polynomial f(x) on V is called a relative invariant if f(p(g} -x) — %((?) 'f(x) for every g^G, with % denoting a rational character of G. It is a homogeneous polynomial and is uniquely determined by % up to a constant factor. Further, (G, p, V) is called regular if it possesses a relative invariant f(x) whose logarithmic Hessian det (92log/(.r) /dx^x^) does not vanish indentically. For example (G, p, V) is known to be regular if G and the isotropy group at a generic point of V are both reductive. And we call it irreducible if p is an irreducible representation. An irreducible regular prehomogeneous vector space has just one irreducible relative invariant up to a constant multiple. Let g be the Lie algebra of G, and x\-^>A-x be the derived repre