Pseudodifferential operators on prehomogeneous vector spaces

Details Pseudodifferential operators on prehomogeneous vector spaces Pseudodifferential operators on prehomogeneous vector spaces

2004-02-09

arxiv.org/abs/math/0402139v1

Mathematics

Representation Theory

27 pages

Scientific paper

Let $G_\C$ be a connected, linear algebraic group defined over $\R$, acting regularly on a finite dimensional vector space $V_\C$ over $\C$ with $\R$-structure $V_\R$. Assume that $V_\C$ posseses a Zariski-dense orbit, so that $(G_\C,\rho,V_\C)$ becomes a prehomogeneous vector space over $\R$. We consider the left regular representation $\pi$ of the group of $\R$-rational points $G_\R$ on the Banach space $\Cvan(V_\R)$ of continuous functions on $V_\R$ vanishing at infinity, and study the convolution operators $\pi(f)$, where $f$ is a rapidly decreasing function on the identity component of $G_\R$. 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 $V_\R-S_\R$ is a smooth operator. Furthermore, if $G_\C$ is reductive, and $S_\C$ and $S_\R$ are irreducible hypersurfaces, $\pi(f)$ corresponds, on each connected component of $V_\R-S_\R$, 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 $V_\R-S_\R$ given by some power $|p(m)|^s$ of a relative invariant $p(m)$ of $(G_\C,\rho,V_\C)$ 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. A trace of $\pi(f)$ can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.

Affiliated with

Also associated with

No associations

Rating

Pseudodifferential operators on prehomogeneous vector spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Pseudodifferential operators on prehomogeneous vector spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Pseudodifferential operators on prehomogeneous vector spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-30392