Pseudodifferential Operators on Prehomogeneous Vector Spaces
Titel:
Pseudodifferential Operators on Prehomogeneous Vector Spaces
Auteur:
Ramacher, Pablo
Verschenen in:
Communications in partial differential equations
Paginering:
Jaargang 31 (2006) nr. 4 pagina's 515-546
Jaar:
2006-01-01
Inhoud:
Let G be a connected, linear algebraic group defined over , acting regularly on a finite dimensional vector space V over with -structure V. Assume that V possesses a Zariski-dense orbit, so that (G, ϱ, V) becomes a prehomogeneous vector space over . We consider the left regular representation π of the group of -rational points G on the Banach space C0(V) of continuous functions on V vanishing at infinity, and study the convolution operators π(f), where f is a rapidly decreasing function on the identity component of G. Denote the complement of the dense orbit by S, and put S = S ∩ V. It turns out that, on V - S, π(f) is a smooth operator. If S = {0}, the restriction of the Schwartz kernel of π(f) to the diagonal defines a homogeneous distribution on V - {0}. Its nonunique extension to V can then be regarded as a trace of π(f). If G is reductive, and S and S are irreducible hypersurfaces, π(f) corresponds, on each connected component of V - S, 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 - S 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, 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.
Tijdschrift beschrijving