The goal of this paper is to describe the relationship between generalized B-opers, generalized SO(2n,C)-opers and (G,P)-opers. In particular, we show that to each generalized B-oper there is a naturally associated (G,P)-oper, but there are some (G,P)-opers that do not arise as generalized B-opers or SO(2n,C)-opers.
Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin’s integrable system and Hecke eigensheaves, 1991), opers have seen several generalizations. In Biswas et al. (SIGMA Symmetry Integr Geom Methods Appl 16:041, 2020), a higher rank analog was studied, named generalized B-opers, where the successive quotients of the oper filtration are allowed to have higher rank and the underlying holomorphic vector bundle is endowed with a bilinear form which is compatible with both the filtration and the oper connection. Since the definition did not encompass the even orthogonal groups, we dedicate this paper to study generalized B-opers whose structure group is SO(2n,C) and show their close relationship with geometric structures on a Riemann surface.
Opers were introduced by Beilinson-Drinfeld [BD93, BD91], and in [Bis03] a higher rank analog was considered, where the successive quotients of the oper filtration are allowed to have higher rank. We dedicate this paper to introducing generalized B-opers (where "B" stands for "bilinear"), obtained by endowing the underlying vector bundle, in the set-up of [Bis03], with a bilinear form which is compatible with both the filtration and the connection. We study the geometry and topology arising through them.