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This is part of the Borel-Weil theorem, more precisely it's Theorem 3 in Serre's 1954 exposition. As he notes there, the algebraicity you ask about is a consequence of Chow's theorem. Also, the orbit of the highest weight space under the compact real form $G_0$ is open in the $G$-orbit by dimension and closed by compactness, so it coincides with the $G$-orbit.