# Cartan subspaces for general algebraic representations

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 $m=2$.

• I'll add the comment here that there are the polar representations of Dadok and Kac, which generalize those studied by Vinberg. – Cheng-Chiang Tsai Apr 15 '16 at 2:14

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).

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 $\mathbb{G}_m\curvearrowright\mathbb{G}_a^{\otimes2}\bigoplus \mathbb{G}_a\bigoplus \mathbb{G}_a^{\otimes-1}$.

• 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? – Jim Humphreys Apr 1 '16 at 13:49
• 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. – Cheng-Chiang Tsai Apr 1 '16 at 14:31
• I tried to build something almost exactly like this definition myself once, without much luck. Is your definition of 'commutes' the same as requiring that the stabiliser of the affine line parallel to $v_2$ through $v_1$ is $\operatorname{Stab}_G(v_1) \cap \operatorname{Stab}_G(v_2)$? – LSpice Dec 12 '19 at 6:29