Envelope of Holomorphy of Matsuki Orbits in Complex Grassmannians

We study the envelopes of holomorphy of Matsuki orbits $M_{\ell,m,r}$ arising as intersections of $G_0$- and $K$-orbits in complex Grassmannians. These orbits, equipped with natural CR-structures, form a class of compact homogeneous CR-submanifolds whose analytic continuation properties are of fundamental interest. Building on Rossi's theory of holomorphic extension, we establish that the envelope of holomorphy of each Matsuki orbit coincides biholomorphically with the containing $K$-orbit $O_{\ell,m}$. Our approach provides a geometric proof based on the holomorphic fiber bundle structure $\pi : M_{\ell,m,r} \to \mathrm{Gr}_{\ell}(E_+) \times \mathrm{Gr}_{m}(E_-)$, clarifying the role of compact isotropic fibers in constraining holomorphic extension. Explicit examples, including the orbit $M_{1,1,1} \subset \mathrm{Gr}_3(\mathbb{C}^8)$, illustrate the method and highlight connections with classical constructions in complex geometry and representation theory.