Generalized zeta integrals on certain real prehomogeneous vector spaces

Abstract

Let $X$ be a real prehomogeneous vector space under a reductive group $G$, such that $X$ is an absolutely spherical $G$-variety with affine open orbit. We define local zeta integrals that involve the integration of Schwartz-Bruhat functions on $X$ against generalized matrix coefficients of admissible representations of $G(\mathbb{R})$, twisted by complex powers of relative invariants. We establish the convergence of these integrals in some range, the meromorphic continuation as well as a functional equation in terms of abstract $\gamma$-factors. This subsumes the Archimedean zeta integrals of Godement-Jacquet, those of Sato-Shintani (in the spherical case), and the previous works of Bopp-Rubenthaler. The proof of functional equations is based on Knop's results on Capelli operators.

arXiv comment37 pages

References in corpus (3)

• Finite multiplicity theorems for induction and restriction
• Zeta integrals, Schwartz spaces and local functional equations
• On the regularity of D-modules generated by relative characters

Cited by in corpus (1)

• Variations on themes of Sato