Une classe d'espaces pr\'ehomog\`enes de type parabolique faiblement sph\'eriques

TL;DR

This paper constructs and analyzes certain prehomogeneous vector spaces associated with specific gradings of simple Lie algebras over local fields, establishing properties of their Zeta functions and functional equations.

Contribution

It introduces new classes of prehomogeneous vector spaces for Lie algebras with specific gradings and studies their Zeta functions and functional equations, generalizing previous results.

Findings

Zeta functions have meromorphic extensions with functional equations.

Explicit formulas for functional equation coefficients and Bernstein polynomials.

Reduction to problems involving centralizers of commuting sl(2) Lie algebras.

Abstract

For absolutely simple, finite-dimensional Lie algebras g of rank at least 2, defined over a local field of characteristic 0 and admitting a graduation: g=g(-2)+g(-1)+g(0)+g(1)+g(2) given by an element H such that 2H is simple, we construct parabolic subgroups P of the automorphism group of g which centralize H, having geometric prehomogeneous action on g(1) and g(-1). We study the structure of these prehomogeneous vector spaces. We prove that the Zeta functions associated to the fundamental invariants for the P action on g(1) and g(-1) have meromorphic extensions which satisfy functional equations. We give the explicit calculus of the coefficients of these functional equations and the Bernstein polynomials associated to these fundamental invariants in the archimedian case, by reducing the problem to a similar problem for centralizers of pair of commuting sl(2) Lie algebras. This work is a generalization of well-known results when g(2)=0.