On functional equations of complex powers
type: Item
instance of: scholarly article
title: On functional equations of complex powers
zbMATH Open document ID: 0599.12017
DOI: 10.1007/BF01388789
author: Jun-ichi Igusa
published in: Inventiones Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143356
review text: Let X be a prehomogeneous vector space (in the sense of Sato-Shintani) over a local field K of characteristic 0, i.e. an affine space together with an irreducible polynomial f over K on X and a connected reductive algebraic K-subgroup \(G\subset GL(X)\) acting transitively on \(Y:=X\setminus f^{-1}(0)\). There exist finitely many \(G_ K\)-orbits \(Y_ 1,...,Y_{\ell}\) making up Y(K); for each \(Y_ i\) and any \(\omega\) in the space \(\Omega\) of all quasi-characters on \(K^{\times}=K\setminus \{0\}\), tempered distributions \(Z_ i(\omega)\) are defined by \[ Z_ i(\omega)(\Phi)=\int_{Y_ i}(\omega \omega_{\kappa})^{-1}(f(x)) \Phi (x) dx \] for Schwartz-Bruhat functions \(\Phi\) on \(X_ K\), where \(\omega_ s(t):=| t|^ s_ K\), \(| \quad |_ K\) denotes valuation in K and \(\kappa:=\dim X/\deg f\). These \(Z_ i\) depend holomorphically on \(\omega\) when restricted to an open subset of \(\Omega\) and admit meromorphic continuations to all of \(\Omega\) with functional equations \[ Z_ i(\omega)^*=\sum^{\ell}_{j=1}\gamma_{ij}(\omega) Z_ j(\omega_{\kappa}\omega^{-1}) \] where * denotes Fourier transform and \(\gamma_{ij}\) are meromorphic on \(\Omega\). The \(\Gamma\)-matrix \((\gamma_{ij}(\omega))\) apparently under the influence of the (polynomial) b-function of f, has been explicitly computed (up to sign) for \(\omega =\omega_ s\), \(K={\mathbb{C}}\) (and reductive G) by earlier authors (e.g. Sato). Now, by introducing a ''renormalized'' \(\Gamma\)-matrix \(a_ K(G,\omega)\) involving \(\gamma\) (\(\omega)\) and the b-function, the author establishes (under the additional restrictions for p-adic fields K that G is irreducible and K-split with the b-function having all its roots in \({\mathbb{Z}})\) that \(a_ K(G,\omega)\) depends only on the K- equivalence class of G and on \(\omega\) (once the ordering of \(Y_ i\) and of similar 'dual orbits' is fixed) and is ''intrinsic'' for \(\ell =1\), in particular. It is explicitly determined and shown to be 1, for \(K={\mathbb{C}}…
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 3966232
zbMATH Keywords: complex powers
zbMATH Keywords: prehomogeneous vector space
zbMATH Keywords: local field
zbMATH Keywords: connected reductive algebraic K-subgroup
zbMATH Keywords: quasi-characters
zbMATH Keywords: tempered distributions
zbMATH Keywords: Schwartz-Bruhat functions
zbMATH Keywords: meromorphic continuations
zbMATH Keywords: functional equations
zbMATH Keywords: \(\Gamma \) -matrix
zbMATH Keywords: b-function
zbMATH Keywords: p-adic fields
zbMATH Keywords: zeta distributions
MaRDI profile type: Publication
cites work: Q5830680
cites work: Sur certains groupes simples
cites work: Sur les multiplicateurs des similitudes
cites work: Zeta functions of simple algebras
cites work: A Classification of Spinors Up to Dimension Twelve
cites work: Some Results on p-Adic Complex Powers
cites work: Theb-functions and holonomy diagrams of irreducible regular prehomogeneous vector spaces
cites work: Algebraic Groups Over Finite Fields
cites work: Les nombres de Tamagawa de certains groupes exceptionnels
cites work: Self-adjoint groups
cites work: On the microlocal structure of the regular prehomogeneous vector space associated with \(SL(5)\times GL(4)\). I
cites work: Q3680914
cites work: On zeta functions associated with prehomogeneous vector spaces
cites work: A classification of irreducible prehomogeneous vector spaces and their relative invariants
cites work: On Dirichlet series whose coefficients are class numbers of integral binary cubic forms
cites work: Q3843673
cites work: Sur certains groupes d'opérateurs unitaires
cites work: Sur la formule de Siegel dans la théorie des groupes classiques
OpenAlex ID: W2081593158
description: scientific article; zbMATH DE number 3966232