Towards Generalized Prehomogeneous Zeta Integrals

Abstract

Let X be a prehomogeneous vector space under a connected reductive group G over R.Assume that the open G-orbit X + admits a finite covering by a symmetric space. We study certain zeta integrals involving (i) Schwartz functions on X, and (ii) generalized matrix coefficients on X + (R) of Casselman-Wallach representations of G(R), upon a twist by complex powers of relative invariants. This merges representation theory with prehomogeneous zeta integrals of Igusa et al. We show their convergence in some shifted cone, and prove their meromorphic continuation via the machinery of bfunction together with V. Ginzburg's results on admissible D-modules. This provides some evidence for a broader theory of zeta integrals associated to affine spherical embeddings.

Zeta Integrals, Schwartz Spaces and Local Functional Equations

“…When restricted to the K × K-finite vectors in V Π , the zeta integrals admit meromorphic continuation by [23, Theorem 15.9.1] or [21,Theorem 8.7]; the latter reference allows general π and F = C. The point is to extend this to all vectors of V Π and obtain meromorphy together with continuity. This has also been achieved in [35]: the basic tool is the method of analytic continuation of Theorem 3.4.5.…”

“…By Theorem 6.2.7, the Axiom 4.4.1 is satisfied in the non-Archimedean case, and it remains to check the properties of Archimedean zeta integrals. It suffices to consider the triple (G, ρ, X) with F = R. The required properties in this case are all established in [35]. Let us give some quick remarks and compare with the original results of Godement-Jacquet.…”

“…• Meromorphic continuation: Godement and Jacquet only considered a K × K-invariant subspace S 0 ⊂ S(X) spanned by "Gaussian × polynomials"; the precise definition depends on ψ as well and can be found in [21, p.115]. This restriction on ξ has been removed in [35]. When restricted to the K × K-finite vectors in V Π , the zeta integrals admit meromorphic continuation by [23, Theorem 15.9.1] or [21,Theorem 8.7]; the latter reference allows general π and F = C. The point is to extend this to all vectors of V Π and obtain meromorphy together with continuity.…”

Generalized zeta integrals on certain real prehomogeneous vector spaces

“…The required holonomicity is furnished by [23], and one can also deduce it from the arguments in [1]. The remaining arguments are the same as in [21].…”

“…We begin by reviewing the basic set-up about prehomogeneous vector spaces from [21]; see also [22,Chapter 6] or [27,14]. The following assumptions will remain in force throughout this article.…”