Zeta-function of a transversely elliptic operator
type: Item
instance of: scholarly article
title: Zeta-function of a transversely elliptic operator
zbMATH Open document ID: 0569.58034
DOI: 10.1007/BF00968952
author: S. A. Smagin
author: M. A. Shubin
published in: Siberian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: In the paper the zeta-function of a transversally elliptic operator is constructed. Let X be a closed, smooth G-manifold, where G is a compact Lie group. Let \(A: C^{\infty}(X,E)\to C^{\infty}(X,E)\) be a classical, pseudodifferential operator of order \(m>0\) and \(T^*_ GX\) the space of covectors conormal to the orbits of the action. It is assumed that for any \(\nu \in T^*_ GX\setminus 0\) and \(\Lambda\in (- \infty,0]\) there exists \((a_ m(\nu)-\Lambda)^{-1}\) where \(a_ m\) denotes the principal symbol of A. Using \(\theta\), an auxiliary symbol of 0-th order with support in a cone neighbourhood of \(T^*_ GX\) and equal to 1 on \(T_ GX\setminus 0\), we are able to construct \(A^ z_{\theta}\) complex power for such an operator, in a way analogous to that for the elliptic operator. Then we define \(\zeta (z)=tr_ GA^ z_{\theta}\) with values in D'(G), where \(<tr_ GB,\eta >=tr BT(\eta)\), \(\eta \in C^{\infty j}(G)\); \(T(\eta)=\int_{G}\eta (g)T(g)dg\) is a bounded operator in \(L^ 2(X,E)\) (T(g) denotes the action of g on sections). \(\zeta\) (z) is holomorphic for \(Re(z)<-\dim X/m\). It is shown in the paper that it extends to the meromorphic function in C with poles of the order \(<\dim X+\dim G\), which lie on the points of arithmetic progression (in R). Moreover the singular part does not depend on the choice of an auxiliary symbol. Namely the following fact is proved: Let \(\theta_ 1\), \(\theta_ 2\) be two auxiliary symbols, then \(\zeta_ 1(z)-\zeta_ 2(z)=tr_ GA^ z_{\theta_ 1}-tr_ GA^ z_{\theta_ 2}\) extends to the holomorphic function in the whole complex plane.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 43A99
zbMATH DE Number: 3909401
zbMATH Keywords: G-vector bundle
zbMATH Keywords: resolution of singularities
zbMATH Keywords: oscillatory integral
zbMATH Keywords: zeta- function
zbMATH Keywords: transversally elliptic operator
reviewed by: Krzysztof P. Wojciechowski
MaRDI profile type: Publication
cites work: Elliptic operators and compact groups
cites work: Representations of compact Lie groups and elliptic operators
cites work: Q3903416
cites work: Complex powers of hypoelliptic pseudo-differential operators with applications
cites work: Q3897569
description: scientific article; zbMATH DE number 3909401