Equivariant index and the moment map for completely integrable torus actions (Q1383907)

The authors consider a completely integrable action of a torus \(T\) on a compact \(\text{Spin}^c\) manifold. The equivariant index of the Dirac operator is then a virtual representation of \(T\). It is shown that the multiplicity of a weight \(\alpha\) equals the winding number around \(\alpha\) of the descended moment map. This number can be any integer, which answers a question posed by R. Bott on the possible values of these multiplicities.