Orbit duality in ind-varieties of maximal generalized flags

We extend Matsuki duality to arbitrary ind-varieties of maximal generalized flags, in other words, to any homogeneous ind-variety $\mathbf {G}/\mathbf {B}$ for a classical ind-group $\mathbf {G}$ and a splitting Borel ind-subgroup $\mathbf {B}\subset \mathbf {G}$. As a first step, we present an explicit combinatorial version of Matsuki duality in the finite-dimensional case, involving an explicit parametrization of $K$- and $G^0$-orbits on $G/B$. After proving Matsuki duality in the infinite-dimensional case, we give necessary and sufficient conditions on a Borel ind-subgroup $\mathbf {B}\subset \mathbf {G}$ for the existence of open and closed $\mathbf {K}$- and $\mathbf {G}^0$-orbits on $\mathbf {G}/\mathbf {B}$, where $\left (\mathbf {K},\mathbf {G}^0\right )$ is an aligned pair of a symmetric ind-subgroup $\mathbf {K}$ and a real form $\mathbf {G}^0$ of $\mathbf {G}$.