A symplectic bundle over an algebraic curve has a natural invariant s Lag determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound on s Lag which is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced by s Lag on moduli spaces of symplectic bundles, and get a full picture for the case of rank fou
