The index of transversally elliptic operators for locally free actions (Q1326969)

Let a connected unimodular Lie group \(G\) act smoothly and locally freely on a closed manifold \(X\). Assume that the isotropy groups of the action are torsion-free. Let \(K\) be the maximal compact subgroup of \(G\). Let \(T\) be a \(G\)-invariant first order differential operator on \(X\) that is ellipitic in directions transverse to the \(G\)-orbits. Using Kasparov products over \(C^* G\), the paper proves index formulas equating indices of elliptic operators on \(K\setminus X\) with linear combinations of multiplicities of \(G\)-representations in \(\text{kernel}(T)- \text{kernel}(T^*)\).