Noncommutative residue. I: Fundamentals.

*(English)*Zbl 0649.58033
\(K\)-theory, arithmetic and geometry, Semin., Moscow Univ. 1984-86, Lect. Notes Math. 1289, 320-399 (1987).

[For the entire collection see Zbl 0621.00010.]

This is the first in a series of papers in which the author will develop a theory of “noncommutative residues” for pseudodifferential operators (\(\psi\) DO’s). This theory has applications to the homological structure of the algebra of \(\psi\) DO’s, to asymptotic expansions and to index formulae.

In this Chapter 1 the author defines the residue in the setting of symplectic geometry. Let \(Y = T^*_ 0 X\) be the cotangent bundle of an n-dimensional manifold \(X\), with the zero section deleted. \(Y\) can be regarded as a principal \(\mathbb{R}^+\)-bundle over the sphere bundle \(Z\) of \(X\). Let omega be the standard symplectic form on \(Y\) and \(Chi\) the vector field which generates the \(\mathbb{R}^+\) action. Define alpha to be the contraction \(i_\chi(\omega)\). Let \(P_{-n}\) be the set of functions on \(Y\) which are homogeneous of degree \(-n\) with respect to the \(\mathbb{R}^+\) action. For \(f\in P_{-n}\) the form \(f(\alpha \wedge (d\alpha)^{n-1})\) can be lifted from a unique form \(\mu_ f\) on \(\mathbb{Z}\). The author defines a map: res\(_ 0: P_ n\to \Omega^{2n - 1}(\mathbb{Z})\) by res\(_ 0(f) = \mu_ f\). For forms with appropriate support the integrated residue is: Res\((f) = \int_ \mathbb{Z} res_ 0(f)\).

In more concrete terms, this can be written as \(\iint_{| xi | = 1} f(x,\xi)dx d \xi'\), where \(d \xi'\) denotes the standard measure on the sphere. Let P* denote the graded Lie algebra of homogeneous functions on Y, under the Poisson bracket. Then (if X is compact, say) the map Res defines a trace on P*, i.e. Res\(\{f,g\} = 0\). The author also defines a ”total residue” mapping the standard chain complex associated to the algebra P* (and the adjoint representation) to the complex of forms on \(\mathbb{Z}\). Let \(A\) be a \(\psi\) DO over X. In local coordinates \(A\) is represented by: \[ As = (1/2\pi)^{n/2}\int p(x,\xi) s(\xi) e^{- ix\xi}d\xi, \] where the matrix value function p has an asymptotic expansion \(p \sim \sum_ r p_{\lambda - r}\), and \(p_\mu\) is homogeneous in \(\xi\) of degree \(\mu\). We put res\((A) = \tau* (res_ 0(p_{-n}))\), a matrix valued function of x. Here \(\tau*\) is the operation of integration over the fibres of \(\mathbb{Z} \to X\). The author shows, using the transformation formulae for \(\psi\) DO’s under coordinated change, that this is an invariant definition, so res(A) is an n-form on \(X\) intrinsically associated to \(A\). (Here we assume for simplicity that \(X\) is oriented.) Suppose \(A\) is an elliptic operator for which the complex powers \(A^{-s}\) can be defined. The (local) \(\zeta\)- function of\(A\), associated to a point \(x \in X\) is the meromorphic continuation of \(\zeta_ x(s) = Tr K_ s(x,x)\), where \(K_ s\) is the kernel of \(A^{-s}\), which is continuous when Re(s) is large. The \(\zeta\)-function has poles at certain values \(s = s_ j\). The author shows that the residues at these points can be expressed in terms of the construction above: res\(_{s = s_ j}(\zeta_ x(s)) = Tr(Res(A^{- s_ j}))\). The \(\zeta\)-function is regular at \(s = 0\) and there is a formula of a similar nature expressing \(\zeta_ x(0)\) in terms of a ”logarithmic symbol” of the operator.

This is the first in a series of papers in which the author will develop a theory of “noncommutative residues” for pseudodifferential operators (\(\psi\) DO’s). This theory has applications to the homological structure of the algebra of \(\psi\) DO’s, to asymptotic expansions and to index formulae.

In this Chapter 1 the author defines the residue in the setting of symplectic geometry. Let \(Y = T^*_ 0 X\) be the cotangent bundle of an n-dimensional manifold \(X\), with the zero section deleted. \(Y\) can be regarded as a principal \(\mathbb{R}^+\)-bundle over the sphere bundle \(Z\) of \(X\). Let omega be the standard symplectic form on \(Y\) and \(Chi\) the vector field which generates the \(\mathbb{R}^+\) action. Define alpha to be the contraction \(i_\chi(\omega)\). Let \(P_{-n}\) be the set of functions on \(Y\) which are homogeneous of degree \(-n\) with respect to the \(\mathbb{R}^+\) action. For \(f\in P_{-n}\) the form \(f(\alpha \wedge (d\alpha)^{n-1})\) can be lifted from a unique form \(\mu_ f\) on \(\mathbb{Z}\). The author defines a map: res\(_ 0: P_ n\to \Omega^{2n - 1}(\mathbb{Z})\) by res\(_ 0(f) = \mu_ f\). For forms with appropriate support the integrated residue is: Res\((f) = \int_ \mathbb{Z} res_ 0(f)\).

In more concrete terms, this can be written as \(\iint_{| xi | = 1} f(x,\xi)dx d \xi'\), where \(d \xi'\) denotes the standard measure on the sphere. Let P* denote the graded Lie algebra of homogeneous functions on Y, under the Poisson bracket. Then (if X is compact, say) the map Res defines a trace on P*, i.e. Res\(\{f,g\} = 0\). The author also defines a ”total residue” mapping the standard chain complex associated to the algebra P* (and the adjoint representation) to the complex of forms on \(\mathbb{Z}\). Let \(A\) be a \(\psi\) DO over X. In local coordinates \(A\) is represented by: \[ As = (1/2\pi)^{n/2}\int p(x,\xi) s(\xi) e^{- ix\xi}d\xi, \] where the matrix value function p has an asymptotic expansion \(p \sim \sum_ r p_{\lambda - r}\), and \(p_\mu\) is homogeneous in \(\xi\) of degree \(\mu\). We put res\((A) = \tau* (res_ 0(p_{-n}))\), a matrix valued function of x. Here \(\tau*\) is the operation of integration over the fibres of \(\mathbb{Z} \to X\). The author shows, using the transformation formulae for \(\psi\) DO’s under coordinated change, that this is an invariant definition, so res(A) is an n-form on \(X\) intrinsically associated to \(A\). (Here we assume for simplicity that \(X\) is oriented.) Suppose \(A\) is an elliptic operator for which the complex powers \(A^{-s}\) can be defined. The (local) \(\zeta\)- function of\(A\), associated to a point \(x \in X\) is the meromorphic continuation of \(\zeta_ x(s) = Tr K_ s(x,x)\), where \(K_ s\) is the kernel of \(A^{-s}\), which is continuous when Re(s) is large. The \(\zeta\)-function has poles at certain values \(s = s_ j\). The author shows that the residues at these points can be expressed in terms of the construction above: res\(_{s = s_ j}(\zeta_ x(s)) = Tr(Res(A^{- s_ j}))\). The \(\zeta\)-function is regular at \(s = 0\) and there is a formula of a similar nature expressing \(\zeta_ x(0)\) in terms of a ”logarithmic symbol” of the operator.

Reviewer: S.Donaldson

##### MSC:

58J40 | Pseudodifferential and Fourier integral operators on manifolds |