Local zeta functions for a class of symmetric spaces

Abstract

This article is a report on a work which will be published elsewhere with complete proofs. It deals with some local zeta functions associated to a family of symmetric spaces arising from 3-gradings of reductive Lie algebras. Let $\tilde{\mathfrak{g}} = {V^{ - }} \oplus \mathfrak{g} \oplus {V^{ + }}$ be a 3-graded real reductive Lie algebra. Let $\tilde{G}$ be the adjoint group of $\tilde{\mathfrak{g}}$ and let G be the analytic subgroup of $\tilde{G}$ corresponding to the Lie algebra g. We make the assumption that the prehomogeneous vector space (G, V ⁺ ) is regular. In our context this means that there exists an irreducible polynomial △₀ on V+ which is relatively invariant under the G-action. Let x₀ be the corresponding character of G. Among the G-orbits in V+ there is the family Ω₀ ⁺,Ω₁ ⁺,…, Ω _r ⁺ of open orbits which are symmetric spaces for G. This means that the isotropy subgroup H _p of an element I _p ⁺ ∈ Ω _p ⁺ is a symmetric subgroup of G. Moreover the same symmetric spaces can be realized on the negative side. The space V ^- has the same number of open G-orbits denoted by Ω₀ ^-, Ω₁ ^-,…, Ω _r ^- and for all p= 0,..., r one has G/ H _p ≃Ω _p ⁺ ≃ Ω _p -. A striking fact in this framework is that all the symmetric spaces G/H _p have the same minimal spherical principal series (π_τ,λ, H _τ,λ) where (τ,λ) is as usual an induction parameter. Let us denote as usual by H _τ,λ ^-∞ the space of the distribution vectors and by (H _τ,λ ^-∞) ^H _p the subspace of H _p ^-fixed vectors. Notice that for a _p ∈(H _τ,λ ^-∞) the function g ↦ π_τ,λ(g)a _p only depends on the class of g in G/Hp and therefore defines a function x ↦ π_τ,λ(g)a _p or a function y ↦π_τ,λ(y)a _p on Ω _p ^- Let S(V ⁺ ) (resp. S (V ^-)) be the space of Schwartz functions on V ⁺ (resp. V-). Then for a = (a ₀, a₁,…, a _r ) ∈П^r _{p= 0}(H _τ,λ ^-∞)^H _p , f ∈ S (V-) we define the following zeta functions:

$$ {Z^{ + }}(f,{\pi _{\tau }}_{{,\lambda }},a) = \sum\limits_{{p = 0}}^{r} {\int_{{\Omega _{p}^{ + }}} {f(x){\pi _{\tau }}_{{,\lambda }}(x)} {a_{p}}d*x,} $$

$$ {Z^{ - }}(h,{\pi _{\tau }}_{{,\lambda }},a) = \sum\limits_{{p = 0}}^{r} {\int_{{\Omega _{p}^{ - }}} {h(y){\pi _{\tau }}_{{,\lambda }}(y)} {a_{p}}d*y,} $$

where d ^* x (resp. d ^* y) is a G-invariant measure on Ω _p ⁺ (resp.Ω _p ^-). We prove that the integrals defining these zeta functions are convergent in a subdomain of the λ parameter, admit meromorphic continuation, and satisfy the following functional equation:

$${{Z}^{ - }}(\mathcal{F}f,{{\pi }_{{\tau ,\lambda }}},a) = {{Z}^{ + }}(f,\chi _{0}^{{ - m}} \otimes {{\pi }_{{\tau ,\lambda }}},{{A}^{{\tau ,\lambda }}}(a)),$$

where F: S(V ⁺)→ S (V-)is the Fourier transform and A ∈ End(П^r _{p= 0} (H _τ,λ ^-∞)^H _p ). Moreover we compute explicitly the matrix A in the standard basis of П^r _{p= 0}(H _τ,λ ^-∞)^H _p . The matrix A generalizes the local Tate “gamma” factor for ℝ.