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 (πτ,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 ↦ πτ,λ(g)a p only depends on the class of g in G/Hp and therefore defines a function x ↦ πτ,λ(g)a p or a function y ↦πτ,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.