Local zeta functions for a class of symmetric spaces

  • Nicole Bopp
  • Hubert Rubenthaler
Part of the Progress in Mathematics book series (PM, volume 220)


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 +1 +,…, 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 , fS (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.


Symmetric Space Zeta Function Parabolic Subgroup Principal Series Open Dense Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Nicole Bopp
    • 1
  • Hubert Rubenthaler
    • 1
  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis Pasteur et CNRSStrasbourg CedexFrance

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