Hecke transformation for orthogonal bundles over curves

Abstract: Given an orthogonal bundle $E$ over a smooth projective curve $X$ we define a Hecke transformation in the moduli space of orthogonal bundles by performing an elementary transformation with respect to a Lagrangian submodule $L \subset E_{2x}$ at some point $x \in X$. We show that the analogue of Tyurin's duality theorem holds for orthogonal bundles. Special cases of orthogonal bundles of ranks $2,3,4$ and $6$ are studied in detail.