The Igusa local zeta function of the simple prehomogeneous vector space \((\text{GL}(1)^4\times \text{SL}(2n+1),\Lambda_2\oplus \Lambda_1\oplus\Lambda_1\oplus\Lambda_1)\)
type: Item
instance of: scholarly article
title: The Igusa local zeta function of the simple prehomogeneous vector space \((\text{GL}(1)^4\times \text{SL}(2n+1),\Lambda_2\oplus \Lambda_1\oplus\Lambda_1\oplus\Lambda_1)\)
zbMATH Open document ID: 1120.11044
author: Satoshi Wakatsuki
published in: Journal of the Mathematical Society of Japan
publication date: 2005-05-03T00:00:00Z
review text: In Theorem 2.1 the author determines explicitly the Igusa local zeta function of the simple prehomogeneous vector space \( \left( \text{GL}(1)\times \text{SL}(2n+1),\Lambda _{2}\oplus \Lambda _{1}\oplus \Lambda _{1}\oplus \Lambda _{1}\right) ,n\geq 1. \) The Igusa zeta function as a distribution on the space of Schwartz-Bruhat functions satisfies a functional equation of the type \(Z(s-\kappa ,\widehat{ \Phi ^{\ast }})=\gamma \left( s\right) Z^{\ast }(-s,\Phi ^{\ast })\), where \( \widehat{\left( \cdot \right) }\) denotes the Fourier transform, \(\Phi ^{\ast }\) is a Schwartz-Bruhat function, and \(Z^{\ast }(s,\Phi ^{\ast })\) is the ''dual'' of \ \(Z(s,\Phi )\). In Theorem 4.1 the author determines explicitly the \(\Gamma \)-factor \(\gamma \left( s\right) \). Furthermore, the author expresses \(\gamma \left( s\right) \) in terms of the Tate local factor and the \(b\)-function of the prehomogeneous space.
reviewed by: W. A. Zúñiga-Galindo
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11S90
zbMATH DE Number: 2164239
zbMATH Keywords: Igusa local zeta function
zbMATH Keywords: prehomogeneous vector spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1160745816
OpenAlex ID: W1991892704
DOI: 10.2969/JMSJ/1160745816
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description: scientific article; zbMATH DE number 2164239