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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.

##### MSC:
 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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