  We reframe Paradan-Vergne's approach to quantization commutes with reduction
in KK-theory through a recent formalism introduced by Kasparov, focusing more
especially the index theoretic parts that lead to their "Witten non-abelian
localization formula". While our method uses the same ingredients as their's in
spirit, interesting conceptual simplifications occur, and the relationship to
the Ma-Tian-Zhang analytic approach becomes quite transparent.
