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In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant
-theory with respect to a compact torus
of various spaces associated to a linear action of
in a vector space
can both be described using some vector spaces of distributions, on the dual of the group
or on the dual of its Lie algebra
. The morphism from
-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a
-transversally elliptic operator on
are determined using the infinitesimal index of the symbol.
In this note, we study an invariant associated with the zeros of the moment map generated by an action form, the infinitesimal index. This construction will be used to study the compactly supported equivariant cohomology of the zeros of the moment map and to give formulas for the multiplicity index map of a transversally elliptic operator.
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