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Yes, the projectivized orbit of a highest weight vector is well known to be the only closed one. It's also an orbit under the compact real form $G_u$, and as such is the unique $G_u$-orbit that is a complex (hence Kähler) submanifold (work of Borel, Weil, Tits, Hirzebruch). In this paper Kostant and Sternberg work out the conditions under which $G_u\cdot[v]$ is a symplectic submanifold. For an exposition see e.g. Guillemin and Sternberg's Symplectic Techniques in Physics.

(Orbits under other real forms are thoroughly studied in J. A. Wolf, The action of a real semisimple group on a complex flag manifold.)

Your problem, however, is that of studying the orbits in the flag manifold $\Bbb P(V)=GL(V)/GL(V)_{[v]}$ P(V)=GL(V)/GL(V)_{[w]}$under any complex (irreducibly represented, reductive)$G\hookrightarrow GL(V)$. In this generality I'm not sure there is much one can say. For example, a theorem of Chevalley (J. Humphreys, Linear algebraic groups, 11.2) asserts that every$G/H$,$H$Zariski closed in$G$, occurs as such an orbit for some (not necessarily irreducible) representation. 4 added 8 characters in body; added 11 characters in body Yes, the projectivized orbit of a highest weight vector is well known to be the only closed one. It's also an orbit under the compact real form$G_u$, and as such is the unique$G_u$-orbit that is a complex (hence Kähler) submanifold (work of Borel, Weil, Tits, Hirzebruch). In this paper Kostant and Sternberg work out the conditions under which$G_u\cdot[v]$is a symplectic submanifold. For an exposition see e.g. Guillemin and Sternberg's Symplectic Techniques in Physics. (Orbits under other real forms are thoroughly studied in J. A. Wolf, The action of a real semisimple group on a complex flag manifold.) Your problem, however, is that of studying the orbits in the flag manifold$\Bbb P(V)=GL(V)/GL(V)_{[v]}$under any complex (irreducibly represented, reductive)$G\hookrightarrow GL(V)$. In this generality I'm not sure there is much one can say. For example, a theorem of Chevalley (J. Humphreys, Linear algebraic groups, 11.2) asserts that every$G/H$,$H$Zariski closed in$G$, occurs as such an orbit for some (not necessarily irreducible) representation. 3 added 531 characters in body; added 7 characters in body Yes, the projectivized orbit of a highest weight vector is well known to be the only closed one. It is It's also an orbit under the only one to be compact real form$G_u$, and as such is the unique$G_u$-orbit that is a complex (hence Kähler) submanifold (work of Borel, Weil, Tits, Hirzebruch). In this paper Kostant and Sternberg work out the conditions under which$G.[v]$G_u\cdot[v]$ is a symplectic submanifold. For an exposition see e.g. Guillemin and Sternberg's Symplectic Techniques in Physics.

The general case is

(Orbits under other real forms are thoroughly studied in J. A. Wolf, The action of a real semisimple group on a complex flag manifold.)

Your problem, however, is that of studying the orbits in the flag manifold $\Bbb P(V)=GL(V)/GL(V)_{[v]}$ under any complex (irreducibly represented) $G\hookrightarrow GL(V)$. In this generality I'm not sure there is much one can say. For example, a theorem of Chevalley (J. Humphreys, Linear algebraic groups, 11.2) asserts that every $G/H$, $H$ closed in $G$, occurs as such an orbit for some (not necessarily irreducible) representation.

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