Abstract
Let G be a torus acting linearly on a complex vector space M and let X be the list of weights of G in M. We determine the topological equivariant K-theory of the open subset M f of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the character lattice \( \widehat{G} \), satisfying the cocircuit difference equations associated to X, introduced by Dahmen and Micchelli in the context of the theory of splines in order to study vector partition functions (cf. [7]).
This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah and Singer [1] which is in a sense universal for index computations.
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Dedicated to Vladimir Morozov on the occasion of his 100th birthday
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De Concini, C., Procesi, C. & Vergne, M. Vector partition functions and index of transversally elliptic operators. Transformation Groups 15, 775–811 (2010). https://doi.org/10.1007/s00031-010-9101-x
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DOI: https://doi.org/10.1007/s00031-010-9101-x