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Let $G$ be a semisimple complex Lie group (or perhaps a reductive algebraic group over $\mathbb{C}$) and $V$ an irreducible finite-dimensional representation of $G$, determined by its highest weight. Is there a complete description of the orbits of $V$, or $\mathbb{P}(V)$, under $G$?

In the case of the adjoint representation of $SL_n$ or $GL_n$, I expect to recover the Jordan canonical form of a matrix, or something of the sort.

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  • $\begingroup$ In the case of the dual of the adjoint representation, this is intensively studied under the keyword "coadjoint orbit." $\endgroup$
    – Qiaochu Yuan
    Jun 19, 2013 at 0:28
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    $\begingroup$ If you want an idea of how hopeless this question is, look at $GL_n$ acting on pairs of matrices, by conjugation. I admit that that's a reducible representation but irreducibility is not going to help much. $\endgroup$
    – Allen Knutson
    Jun 19, 2013 at 1:00
  • $\begingroup$ Allen's cautionary example is a good one. Another example, one that illustrates the complexity by relating it to something geometric, is to look at the (irreducible) representation of $\mathrm{GL}(3,\mathbb{C})$ on the homogeneous polynomials of degree $d$ in $3$ variables. The orbits in this case are essentially the projective equivalence classes of plane curves of degree $d$, which is known to be an enormously complex zoo, with no hope of a normal form. Even the very classical case $d=3$ is nontrivial. $\endgroup$
    – Robert Bryant
    Jun 20, 2013 at 12:06

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As Allen has indicated, there is no hope to solve this question even for $G=GL_n.$ However, there are some representations that are mild enough to be analogous to the vector (or standard) representation of $GL_n$ on $\mathbb{C}^n$ (all non-zero vectors belong to the same orbit) or to the adjoint representation of $GL_n$, where the classification problem is equivalent to determining the conjugacy classes of matrices. The following classes of "well-behaved" representations, listed in the order of increasing complexity, have been extensively studied: multiplicity-free actions, prehomogeneous vector spaces, Kostant--Rallis actions. In particular, $\mathbb{C}^n, S^{2}\mathbb{C}^n$ and $\Lambda{^2}\mathbb{C}^n$ are multiplicity-free and $M_{n,n}$ is a Kostant--Rallis action. For all these classes, the invariant algebra is either trivial (scalars) or a free commutative algebra. Unfortunately, most representations fall outside of these classes, even if you restrict attention to the irreducible polynomial representations of $GL_n.$

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