Pseudodifferential operators on prehomogeneous vector spaces

Abstract

Let $G_\C$ be a connected, linear algebraic group defined over $\mathbb{R}$, acting regularly on a finite dimensional vector space $V_\C$ over $\C$ with $\mathbb{R}$-structure $V_{\mathbb{R}}$. Assume that $V_\C$ posseses a Zariski-dense orbit, so that $(G_\C,ρ,V_\C)$ becomes a prehomogeneous vector space over $\mathbb{R}$. We consider the left regular representation $\pi$ of the group of $\mathbb{R}$-rational points $G_{\mathbb{R}}$ on the Banach space $\Cvan(V_\R)$ of continuous functions on $V_{\mathbb{R}}$ vanishing at infinity, and study the convolution operators $\pi (f)$, where $f$ is a rapidly decreasing function on the identity component of $G_{\mathbb{R}}$. Denote the complement of the dense orbit by $S_\C$, and put $S_\R=S_\C\cap V_\R$. It turns out that the restriction of $\pi (f)$ to $V_{\mathbb{R}}-S_{\mathbb{R}}$ is a smooth operator. Furthermore, if $G_\C$ is reductive, and $S_\C$ and $S_{\mathbb{R}}$ are irreducible hypersurfaces, $\pi (f)$ corresponds, on each connected component of $V_{\mathbb{R}}-S_{\mathbb{R}}$, to a totally characteristic pseudodifferential operator. We then investigate the restriction of the Schwartz kernel of $\pi (f)$ to the diagonal. It defines a distribution on $V_{\mathbb{R}}-S_{\mathbb{R}}$ given by some power $|p(m){|}^s$ of a relative invariant $p(m)$ of $(G_\C,ρ,V_\C)$ and, as a consequence of the fundamental theorem of prehomogeneous vector spaces, its extension to $V_{\mathbb{R}}$, and the complex $s$-plane, satisfies functional equations. A trace of $\pi (f)$ can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.

27 pages