Equivariant index of twisted Dirac operators and semi-classical limits (Q2417744)

Let \(G\) be a compact connected Lie group with Lie algebra \(g\). Let \(M\) be a compact spin manifold with a \(G\)-action, and \(L\) be a \(G\)-equivariant line bundle on \(M\). Consider an integer \(k\), and let \(\operatorname{Ind}_G(M,L^k)\) be the equivariant index of the Dirac operator on \(M\) twisted by \(L^ k\). Consider the geometric (rescaled) analogue of \(\operatorname{Ind}_G(M,L^k)\). The authors prove that this geometric (rescaled) analogue had an asymptotic expansion when \(k\) is large. If \(M\) is noncompact, they use these asymptotic techniques to give another proof of the fact that the formal geometric quantization of a manifold with a spinc structure is functorial with respect to restriction to subgroups. For the entire collection see [Zbl 1412.22001].