We define an equivariant index of -Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. Our main result is that the index decomposes into irreducible representations according to the quantization commutes with reduction principle.
"Equivariant indices of -Dirac operators for proper moment maps." Duke Math. J. 166 (6) 1125 - 1178, 15 April 2017. https://doi.org/10.1215/00127094-3792923