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Asymptotics and index for families invariant with respect to a bundle of Lie groups. (English) Zbl 1048.58018

Summary: In another paper [Acta Math. Hung. 99, No. 1–2, 155–183 (2003; Zbl 1026.19007)] we defined the gauge-invariant index of a family of elliptic operators invariant with respect to the free action of a bundle \({\mathcal G}\to B\) of Lie groups and we proved an index formula when the fibers of \({\mathcal G}\to B\) are simply-connected solvable groups.
In this paper, we study traces on the corresponding algebras of invariant families of pseudodifferential operators and we obtain a local index formula using the Fedosov product. For topologically non-trivial bundles we have to use methods of non-commutative geometry. We then discuss, as an application, the construction of higher eta-invariants, which are morphisms \(K_n(\Psi^\infty_{\text{inv}}(Y))\to{\mathbb C}\). The algebras of invariant pseudodifferential operators that we study, \(\psi^\infty_{\text{inv}}(Y)\) and \(\Psi^\infty_{\text{inv}}(Y)\), are generalizations of parameter-dependent algebras of pseudodifferential operators (with parameter in \({\mathbb R}^q\)), so our results also provide asymptotics and an index theorem for elliptic, parameter-dependent pseudodifferential operators.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
58J40 Pseudodifferential and Fourier integral operators on manifolds

Citations:

Zbl 1026.19007
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