Counting maximal isotropic subbundles of orthogonal bundles over a curve

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Let C be a smooth projective curve and V an orthogonal bundle over C. Let IQe(V) be the isotropic Quot scheme parameterizing degree e isotropic subsheaves of maximal rank in V. We give a closed formula for intersection numbers on components of IQe(V) whose generic element is saturated. As a special case, for g≥2, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in [16], and of the symplectic case studied in [7].