Moduli spaces of vector bundles on a curve and opers

Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \,2$$ , and let $$\xi $$ holomorphic line bundle on ^{\otimes 2}\,=\, {\mathcal O}_X$$ . Fix theta characteristic $${\mathbb {L}}$$ X. $${\mathcal M}_X(r,\xi )$$ the moduli space stable vector bundles E rank r such that $$\bigwedge ^r E\,=\, \xi $$H^0(X,\, E\otimes {\mathbb L})\,=\, 0$$ Consider quotient by involution given $$E\, \longmapsto \, E^*$$ We construct an algebraic morphism from this to $$\textrm{SL}(r,{\mathbb C})$$ opers Since $$\dim coincides dimension opers, it is natural ask about injectivity surjectivity map.