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(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 γ-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. MSC (2010) 11S40; 11S90 43A85 KeywordsZeta integrals, prehomogeneous vector spaces, Capelli operators where η ∈ N π (X + ), v ∈ V π , and ξ ∈ S(X). The goal of this article is to prove three basic properties of these integrals, in increasing level of difficulty:Convergence (Theorem 3.10) The integral Z λ (η, v, ξ) converges for Re(λ) ≥ X κ for some κ ∈ Λ R depending only on π and (G, ρ, X), and it is jointly continuous in (v, ξ) in that range.Meromorphic continuation (Theorem 3.12) Z λ (η, v, ξ) admits a meromorphic continuation to all λ ∈ Λ C . To be precise, there exists a holomorphic function L(η, λ) on Λ C for any given η, not identically zero, such that LZ λ (η, v, ξ) := L(η, λ)Z λ (η, v, ξ) extends holomorphically to all λ ∈ Λ C .Functional equation (Theorem 3.13) Fix an additive character ψ and denote the integral for (G, ρ, X) as Žλ . There is then a unique meromorphic family of C-linear maps γ(π, λ) :for all η ∈ N π ( X+ ), v ∈ V π and ξ ∈ S(X), where both sides are viewed as meromorphic families in λ ∈ Λ C .Moreover, one can obtain slightly more information on the "denominator" L(η, λ), and describe the dependence of γ(λ, π) on ψ; it turns out that the γ-factor, which is actually a linear transform, is generically invertible (Proposition 3.14). We refer to the cited Theorems for the precise statements.Note that our formalism is non-trivial only when N π (X + ) = {0}; in other words, π must be distinguished by X + (R). Distinguished representations and their generalized matrix coefficients are the main concerns of harmonic analysis on spherical varieties.The same result hold for prehomogeneous vector spaces over C; see §3.4.