Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $mathfrak{osp}_{2m+1|2n}$ and $mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${}_m$ corresponding to $mathfrak{osp}_{2m|0}=mathfrak{so}_{2m}$.