# A positive characteristic theory for polar representations?

Let $G$ be a split reductive algebraic group over a field $k$ of characteristic zero and $\mathfrak{g}$ it's Lie algebra. If $T\subset G$ is a maximal torus with Lie algebra $\mathfrak{t}$ and Weyl group $W$, then there is a well-known isomorphism of algebras
$$k[\gfr]^G\xrightarrow{\sim}k[\mathfrak{t}]^W.$$ This is called the Chevalley restriction theorem. There are many ways to generalize this. For example, consider the same setup but now take an automorphism $\theta:G\to G$ such that $\theta^2 = 1$. Then the fixed point group $G_0 = G^\theta$ now acts on
$$\mathfrak{g}_1 = \{ x\in \mathfrak{g} : \theta(x) = -x\}.$$ Here, we have abused notation and written $\theta$ for the derivative of $\theta:G\to G$ acting on $\mathfrak{g}$. This type of representation is known as a Vinberg $\theta$-group, even though its actually a representation. Again, there is a notion of a Cartan subspace $\mathfrak{a}\subseteq\mathfrak{g}$, a "little" Weyl group $W$, and there is an isomorphism
$$k[\mathfrak{g}]^{G_0}\xrightarrow{\sim}k[\mathfrak{a}]^W.$$ Both the usually Chevalley restriction theorem and this version for an involution can actually be generalized to a field $k$ with sufficiently large positive characteristic. The case for involutions, and indeed more general finite order automorphisms in positive characteristc has been solved by Levy [1].

Chevalley's theorem has been generalized also to many other interesting cases. One of them is the case of polar representations, defined by Dadok and Kac [2]. A polar representation is a kind of generalization of the previous case of Lie algebras with automorphism. In this case, we have a reductive algebraic group $G$ acting linearly on a vector space $V$ over the complex numbers. One can define the following space for any $v\in V$ with closed $G$-orbit:
$$c_v = \{ x\in V : \mathfrak{g}x\subseteq \mathfrak{g}v \}.$$ Here, $\mathfrak{g}$ is the Lie algebra of $G$ as usual. If it happens that $\dim c_v = \dim \C[V]^G$, then we say that $c_v$ is a Cartan subspace.

I bet you can guess where this is going: we want to find an isomorphism between $\C[V]^G$ and something else, probably something like $\C[c_v]^W$ where $W$ is some kind of Weyl group. Now, if we can find such a $v\in V$ which is semisimple, then $c_v$ is called a Cartan subspace and the representation is called polar. We can also define its Weyl group in the usual way: $W = N_G(c_v)/Z_G(c_v)$. That is, the normalizer modulo the centralizer of $c_v$. It's no surprise that:

Theorem. [Dadok and Kac] Let $V$ be a polar representation of $G$ and let $c_v$ be a Cartan subspace with Weyl group $W$. Then $\mathbb{C}[V]^G\cong \mathbb{C}[c_v]^W$.

Notice that this Chevalley restriction theorem for polar representations is stated over the complex numbers. Unlike Levy's excellent work generalizing Vinberg's theory to positive characteristic, there is no analogous generalization to arbitrary polar representations in positive characteristic. Part of the problem here is that many of the definitions and results rely crucially on characteristic zero. In the case of Vinberg's $\theta$-groups where there is extra structure, one just has to read Levy's papers to see the enormous amount of extra work needed to handle sufficiently large positive characteristic.

I'm not sure if we'll see a positive characteristic generalization (and it's not something I'm working on at the moment) but I thought I'd put this out there as it is a problem I've thought about for a while in the past. If such a generalization were possible, it might be quite useful in the theory of automorphic representations.

## References

1. Levy, Paul. Vinberg's $\theta$-groups in positive characteristic and Kostant-Weierstrass slices. Transform. Groups 14 (2009), no. 2, 417-461.
2. Dadok, Jiri; Kac, Victor. Polar representations. J. Algebra 92 (1985), no. 2, 504-524