Biquantization of symmetric pairs and the quantum shift

The biquantization of symmetric pairs was studied by Cattaneo et al. (2008)  [1] in terms of Kontsevich-like graphs. This note, also in view of recent results by Calaque et al. (2009)  [3], amends a minor mistake that did not spoil the main results of the paper. The mistake consisted in ignoring a regular term in the boundary contribution of some propagators. On the other hand, its correction brings back the quantum shift, present in the approaches by the orbit method, that was otherwise puzzlingly missing. In addition a detailed comparison of the two, equivalent, ways of defining biquantization working on the upper half plane or on one quadrant is presented, as well as a more conceptual approach to biquantization and the due corrections of some results of Cattaneo et al. (2008)  [1] in view of the aforementioned correction by the quantum shift.