Geodesic
Formal equivariant $\widehat A$ class, splines and multiplicities of the index of transversally elliptic operators
Izvestiya. Mathematics , Tome 80 (2016) no. 5, pp. 958-993.

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Let $G$ be a connected compact Lie group acting on a manifold $M$ and let $D$ be a transversally elliptic operator on $M$. The multiplicity of the index of $D$ is a function on the set $\widehat G$ of irreducible representations of $G$. Let $T$ be a maximal torus of $G$ with Lie algebra $\mathfrak t$. We construct a finite number of piecewise polynomial functions on $\mathfrak t^*$, and give a formula for the multiplicity in terms of these functions. The main new concept is the formal equivariant $\widehat A$ class.
Keywords: equivariant $K$-theory, splines.
Mots-clés : equivariant index 
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M. Vergne. Formal equivariant $\widehat A$ class, splines and multiplicities of the index of transversally elliptic operators. Izvestiya. Mathematics , Tome 80 (2016) no. 5, pp. 958-993. http://geodesic.mathdoc.fr/item/IM2_2016_80_5_a7/

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