Isotropic Quot schemes of orthogonal bundles over a curve

We study the isotropic Quot schemes $I{Q}_e(V)$ parameterizing degree $e$ isotropic subsheaves of maximal rank of an orthogonal bundle $V$ over a curve. The scheme $I{Q}_e(V)$ contains a compactification of the space $I{Q}_e^o(V)$ of degree $e$ maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [6]. We observe that for certain topological types of $V$, the scheme $I{Q}_e(V)$ is empty for all $e$. In the remaining cases, for infinitely many $e$ there are irreducible components of $I{Q}_e(V)$ consisting entirely of nonsaturated subsheaves, and so $I{Q}_e(V)$ is strictly larger than the closure of $I{Q}_e^o(V)$. As our main result, we prove that for any orthogonal bundle $V$ and for $e\ll 0$, the closure $\overline{I{Q}_e^o(V)}$ of $I{Q}_e^o(V)$ is either empty or consists of one or two irreducible connected components, depending on $\mathrm{deg}(V)$ and $e$. In so doing, we also characterize the nonsaturated part of $\overline{I{Q}_e^o(V)}$ when $V$ has even rank.