Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a maximal isotropic subbundle. This invariant and the induced stratifications on moduli spaces of orthogonal bundles were studied for bundles of even rank in [I. Choe and G. H. Hitching, A stratification on the moduli space of symplectic and orthogonal bundles over a curve, Internat. J. Math. 25(5) (2014), Article ID: 1450047, 27pp.]. In this paper, we obtain analogous results for bundles of odd rank. We compute the sharp upper bound on the isotropic Segre invariant. Also we show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions.