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The Chern character of a transversally elliptic symbol and the equivariant index. (English) Zbl 0847.46037
Let \(G\) be a compact Lie group acting on a compact manifold \(M\) with the cotangent bundle \(T^* M\), \(T^*_G M\) be the union of the spaces \((T^*_G M)_x\), \(x\in M\), where \((T^*_G M)_x \subset (T^*_x M)\) is the orthogonal of the tangent space at \(x\) to the orbit \(G(x)\). The \(G\)-equivariant index of the \(G\)-transversally elliptic pseudodifferential operator \(P\) on \(M\) is the generalized function on \(G\) defined by \[ \text{index}^G(P)(g):= \text{Tr}(g, \text{Ker } P)- \text{Tr}(g, \text{Ker } P^*). \] The principal symbol \(\sigma(P)\) defines an element \([\sigma(P)]\) of \(K_G(T^*_G M)\). Moreover, the \(G\)-equivariant index of \(P\) depends only on the class \([\sigma(P)]\in K_G(T^*_G M)\). Thus, the \(G\)-equivariant index induces a homomorphism of \(R(G)\)-modules \[ \text{index}^{G, M}_a: K_G(T^*_G M)\to C^{- \infty}(G)^G \] which is called the analytical index.
The purpose of this article is to define the cohomological index \[ \text{index}^{G, M}_c: K_G (T^*_G M) \to C^{- \infty} (G)^G, \] by means of the bouquet integral of a certain family of equivariant differential forms on the cotangent bundle \(T^* M\).
The purpose of the next article [the authors, Invent. Math. 124, 51-101 (1996)] is to prove that these two maps – the analytical and the cohomological index – coincide.

46L80 \(K\)-theory and operator algebras (including cyclic theory)
47G30 Pseudodifferential operators
58J20 Index theory and related fixed-point theorems on manifolds
19K56 Index theory
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