Biquantization of symmetric pairs and the quantum shift

TL;DR

This paper corrects a minor mistake in the biquantization of symmetric pairs, restoring the quantum shift and providing a detailed comparison of different approaches, with implications for character theory and boundary contributions.

Contribution

It amends a small error in previous biquantization work, reintroduces the quantum shift, and compares different geometric approaches with a conceptual framework.

Findings

Restoration of the quantum shift in biquantization

Comparison of upper half plane and quadrant approaches

Corrections to previous results on biquantization and triquantization

Abstract

The biquantization of symmetric pairs was studied in Cattaneo and Torossian in terms of Kontsevich-like graphs. This paper, also in view of recent results in Calaque et al, 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 and Torossian in view of the aforementioned correction by the quantum shift. Finally, we review the triquantization construction developed for the treatment of characters by taking into accounts the same boundary problem as for the biquantization.