Pseudodifferential Operators on Prehomogeneous Vector Spaces

Let G be a connected, linear algebraic group defined over ℝ, acting regularly on a finite dimensional vector space V over ℂ with ℝ-structure V ℝ . Assume that V possesses a Zariski-dense orbit, so that (G, ϱ, V) becomes a prehomogeneous vector space over ℝ. We consider the left regular representation π of the group of ℝ-rational points G ℝ on the Banach space C 0 (V ℝ ) of continuous functions on V ℝ vanishing at infinity, and study the convolution operators π(f), where f is a rapidly decreasing function on the identity component of G ℝ . Denote the complement of the dense orbit by S, and put S ℝ = S ∩ V ℝ . It turns out that, on V ℝ − S ℝ , π(f) is a smooth operator. If S ℝ = {0}, the restriction of the Schwartz kernel of π(f) to the diagonal defines a homogeneous distribution on V ℝ − {0}. Its nonunique extension to V ℝ can then be regarded as a trace of π(f). If G is reductive, and S and S ℝ are irreducible hypersurfaces, π(f) corresponds, on each connected component of V ℝ − S ℝ , to a totally characteristic pseudodifferential operator. In this case, the restriction of the Schwartz kernel of π(f) to the diagonal defines a distribution on V ℝ − S ℝ given by some power |p(m)| s of a relative invariant p(m) of (G, ϱ, V) and, as a consequence of the Fundamental Theorem of Prehomogeneous Vector Spaces, its extension to V ℝ , and the complex s-plane, satisfies functional equations similar to those for local zeta functions. A trace of π(f) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.