Zeta-function of a transversely elliptic operator
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.
scientific article; zbMATH DE number 3909401