Pseudodifferential Operators on Prehomogeneous Vector Spaces

Abstract

Abstract. 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 , ̺, V C ) 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 C , and put S R = S C ∩ V R . It turns out that the restriction of π(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, π(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 π(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 , ̺, 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 π(f ) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.

Microlocal analysis on wonderful varieties. Regularized traces and global characters

“…This paper is based on the local structure theorem for real loci of strict wonderful G-varieties recently proved by Akhiezer and Cupit-Foutou [1], and generalizes results already obtained by Parthasarathy and Ramacher [17] for the Oshima compactification of a Riemannian symmetric space, as well as earlier work of Ramacher [18]. Actually, the mentioned local structure theorem provides a natural framework in which the previous results can be understood in a conceptually simple way.…”

“…Besides, it should be noticed that the space is different from the Schwartz space introduced by Harish–Chandra . In our context, the introduction of the space was motivated by the study of strongly elliptic operators and the decay properties of the semigroups generated by them .…”

“…Remark The proof of Proposition is modelled on the proof of [, Theorem 4] and [, Theorem 2], where the integral transform was not explicitly introduced yet, but is tacitly present.…”

Equivariant heat asymptotics on spaces of automorphic forms

“…being at most of exponential growth in g. Let next f 1 ∈ S(G), and assume that f 2 ∈ C ∞ (G), together with all its derivatives, is at most of exponential growth. Then, by [20], Proposition 1, we have…”

“…Proof. Our considerations will essentially follow the proof of Theorem 4 in [20], or Theorem 2 in [19]. For a review on pseudodifferential operators, the reader is referred to [23].…”