Parametrizing maximal compact subvarieties

Abstract:

For the Lie group $G = \mathrm {Sp}(n, \mathbb {R} )$, let $D_i$ be the open $G-$orbit of Lagrangian planes of signature $(i,n-i)$ in the generalized flag variety of Lagrangian planes in $\mathbb {C} ^{2n}$. For a suitably chosen maximal compact subgroup $K$ of $G$ and a base point $x_i$ we have that the $K-$orbit of $x_i$ is a maximal compact subvariety of $D_i$. We show that for $i = 1, \dots , n-1$ the connected component containing $Kx_i$ in the space of $G_{\mathbb {C}}$ translates of $Kx_i$ which lie in $D_i$ is biholomorphic to $G/K \times {\overline {G/K}}$, where ${\overline {G/K}}$ denotes $G/K$ with the opposite complex structure.