Local Zeta Functions Attached To The Minimal Spherical Series For A Class Of Symmetric Spaces
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Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
Intends to prove a functional equation for a local zeta function attached to the minimal spherical series for a class of real reductive symmetric spaces.
Local Zeta Functions Attached to the Minimal Spherical Series for a Class of Symmetric Spaces
The aim of this paper is to prove a functional equation for a local zeta function attached to the minimal spherical series for a class of real reductive symmetric spaces. These symmetric spaces are obtained as follows. We consider a graded simple real Lie algebra $\widetilde{\mathfrak g}$ of the form $\widetilde{\mathfrak g}=V^-\oplus \mathfrak g\oplus V^+$, where $[\mathfrak g,V^+]\subset V^+$, $[\mathfrak g,V^-]\subset V^-$ and $[V^-,V^+]\subset \mathfrak g$. If the graded algebra is regular, then a suitable group $G$ with Lie algebra $\mathfrak g$ has a finite number of open orbits in $V^+$, each of them is a realization of a symmetric space $G\slash H_p$.The functional equation gives a matrix relation between the local zeta functions associated to $H_p$-invariant distributions vectors for the same minimal spherical representation of $G$. This is a generalization of the functional equation obtained by Godement} and Jacquet for the local zeta function attached to a coefficient of a representation of $GL(n,\mathbb R)$.
Quasi-Ordinary Power Series and Their Zeta Functions
Author: Enrique Artal-Bartolo
language: en
Publisher: American Mathematical Soc.
Release Date: 2005
Intends to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, this title computes the local Denef-Loeser motivic zeta function $Z_{\text{DL}}(h, T)$ of a quasi-ordinary power series $h$ of arbitrary dimension