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So I feel like asking the following likely open-ended question: What good generalizations of the notion of Cartan subspace do we have?

To be precise, let $G\curvearrowright V$ be an algebraic representation over a field $k$ satisfying some conditions you like (e.g. $\mathrm{char}(k)=0$, $G$ is connected reductive). When can we call a subspace $S\subset V$ a Cartan subspace with a good reason? (hopefully generalizing the notion of Vinberg as duplicated below, at least that of Cartan subalgebras for adjoint representations for sure)

My motivation is that I am trying to generalize some work of myself about affine Springer fibers and possibly a bit Springer theory. I end up deciding that the notion of Cartan subspace is essential.

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When $\mathrm{char}(k)=0$, $H$ is a reductive group, $\theta$ is an automorphism on $H$ of finite order $m$, $\zeta_m\in k$ is a primitive $m$-th root of unity, $G:=(H^{\theta})^o$, $V:=\mathfrak{h}^{\theta=\zeta_m}$ and $G\curvearrowright V$ comes from the original adjoint representation $H\curvearrowright\mathfrak{h}$, Vinberg defined a Cartan subspace to be a maximal subspace consisting of commuting semisimple elements in $V$. Here commuting means commuting in $\mathfrak{h}$, and semisimple means its $G$-orbit is closed, or equivalently it is semisimple in $\mathfrak{h}$. Vinberg proved that all Cartan subspaces are conjugate, generalizing previous results for symmetric spaces when $m=2$.

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  • $\begingroup$ I'll add the comment here that there are the polar representations of Dadok and Kac, which generalize those studied by Vinberg. $\endgroup$ – Cheng-Chiang Tsai Apr 15 '16 at 2:14
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Assume $\text{char}\,k=0$ for simplicity. Then there are two further generalizations of Cartan subspaces which come to my mind:

  1. A linear subspace $S\subseteq V$ such that the restricted quotient morphism $S\to V/\!\!/G$ is finite and surjective. Such a thing exists if and only if the quotient morphism $\pi:V\to V/\!\!/G$ is equidimensional or, equivalently, the nullcone $\pi^{-1}(0)$ has the minimal possible dimension, namely $\dim V-\dim V/\!\!/G$. There are classifications for $G$ simple (Schwarz) or $V$ irreducible (Littelmann).
  2. A generalization of Chevalley's restriction theorem due to Luna-Richardson. For that let $H\subseteq G$ be the principal stabilizer, i.e., $H$ is conjugate to the stabilizer of a generic closed orbit. Let $S:=V^H$ be the fixed point set. Then $S/\!\!/N_G(H)\to V/\!\!/G$ is an isomorphism. For the adjoint representation, $H$ is a maximal torus, $S$ is a Cartan subspace and the action of $N_G(H)$ on $S$ factors through the Weyl group. So one gets exactly Chevalley's theorem.

There is also a nice discussion of so-called sections in a survey on Invariant Theory by Popov-Vinberg (Encyclopedia of Math. Sci. Vol 55).

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This is an (not satisfying) attempt of myself.

Say a vector $v\in V$ is semisimple if its $G$-orbit is closed. Say two vectors $v_1,v_2\in V$ commute if there exists $c\in\bar{k}^{\times}$ such that $\mathrm{Stab}_G(v_1+cv_2)=\mathrm{Stab}_G(v_1)\cap\mathrm{Stab}_G(v_2)$. Define a Cartan subspace to be a subspace $S\subset V$ of commuting semisimple elements such that the map $S\hookrightarrow V\twoheadrightarrow V/\!/G$ is finite and surjective.

Such Cartan subspaces aren't all conjugate for $\mathrm{SL}_3\curvearrowright\mathrm{Sym}^2(std)\bigoplus\mathrm{Sym}^2(std)$ (though for my own purpose it's fine here). They don't exist for $\mathbb{G}_m\curvearrowright\mathbb{G}_a^{\otimes2}\bigoplus \mathbb{G}_a\bigoplus \mathbb{G}_a^{\otimes-1}$.

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  • $\begingroup$ The question you've formulated does look open-ended though worthwhile. Maybe it's simplest just to add this "answer" to the original question, as a remark? $\endgroup$ – Jim Humphreys Apr 1 '16 at 13:49
  • $\begingroup$ Thanks. I wish for a [hide][/hide]-mode... I felt like my attempt does not really deserve to be in the question, so maybe it still make sense not to distract people with this after we have Vinberg example. $\endgroup$ – Cheng-Chiang Tsai Apr 1 '16 at 14:31

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