On the index of nonlocal elliptic operators for compact Lie groups. (English) Zbl 1248.58013
Let a compact Lie group \(G\) act on a closed smooth manifold \(M\). B. Yu. Sternin [Cent. Eur. J. Math. 9, No. 4, 814–832 (2011; Zbl 1241.58012)] defined the notion of ellipticity for a nonlocal operator \(D\) of the form \(1 + \int _{G} D_{g} \circ T_{g} \, dg\) and showed that such elliptic operators are Fredholm. Here \(T_{g}\) represents the translation action of \(G\) on a space of functions on \(M\), and \(\{ D_{g} : g\in G\}\) is a smooth family of pseudodifferential operators on \(M\). To define ellipticity, Sternin lifted \(D\) to an operator \(\widehat{D}\) on \(M \times G\), defined the symbol of \(\widehat{D}\) as a function with values in bounded operators on \(L^{2}(G)\), and let ellipticity mean that this symbol was transversally elliptic in the sense that its values were invertible on the zero section’s complement in the transverse cotangent bundle. The Fredholm index of \(D\) can be identified with the \(G\)-invariants of the index of this transversally elliptic operator \(\widehat{D}\).
The paper under review gives a formula for this Fredholm index in the case that \(G\)’s action is locally free. This index formula recovers the results of V. E. Nazaikinskii, A. Yu. Savin and B. Yu. Sternin [Elliptic theory and noncommutative geometry. Nonlocal elliptic operators. Basel: Birkhäuser (2008; Zbl 1158.58013)] in the case that \(G\) is finite. Among the ingredients in the general compact \(G\) case is a connection on the \(L^{2}(G)\) bundle that can be used to define a Chern character with values in basic cohomology. Using an increasing sequence of \(G\)-invariant subspaces of \(L^{2}(G)\) (actually of vector-valued \(L^{2}\) functions on \(G\)), the author realizes his transversally elliptic operator as the norm limit of a sequence of transversally elliptic operators acting on sections of finite-dimensional bundles. To these the author applies the index theorems of T. Kawasaki [Nagoya Math. J. 84, 135–157 (1981; Zbl 0437.58020)] and M. Vergne [Duke Math. J. 82, No. 3, 637–652 (1996; Zbl 0874.57029)]. Finally the author exhibits more directly an expression of the index formula in terms of the original operator \(D\).
The paper under review gives a formula for this Fredholm index in the case that \(G\)’s action is locally free. This index formula recovers the results of V. E. Nazaikinskii, A. Yu. Savin and B. Yu. Sternin [Elliptic theory and noncommutative geometry. Nonlocal elliptic operators. Basel: Birkhäuser (2008; Zbl 1158.58013)] in the case that \(G\) is finite. Among the ingredients in the general compact \(G\) case is a connection on the \(L^{2}(G)\) bundle that can be used to define a Chern character with values in basic cohomology. Using an increasing sequence of \(G\)-invariant subspaces of \(L^{2}(G)\) (actually of vector-valued \(L^{2}\) functions on \(G\)), the author realizes his transversally elliptic operator as the norm limit of a sequence of transversally elliptic operators acting on sections of finite-dimensional bundles. To these the author applies the index theorems of T. Kawasaki [Nagoya Math. J. 84, 135–157 (1981; Zbl 0437.58020)] and M. Vergne [Duke Math. J. 82, No. 3, 637–652 (1996; Zbl 0874.57029)]. Finally the author exhibits more directly an expression of the index formula in terms of the original operator \(D\).
Reviewer: Peter Haskell (Blacksburg)
MSC:
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 46L87 | Noncommutative differential geometry |
| 19K56 | Index theory |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
Keywords:
nonlocal operator; basic cohomology; crossed product; transverse ellipticity; Chern character; index formula; Fredholm indexReferences:
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