Non-linear equivariant maps between group representations Given two representations $\pi_1$ and $\pi_2$ of a group $G$ (let's say it's a compact Lie group), a natural thing to study are linear equivariant maps A between them:
$$
A \pi_1 = \pi_2 A
$$
I'm wondering if it is possible to study non-linear (e.g. polynomial) equivariant maps beween representations.
To give a concrete example, consider the following representations of $\operatorname{SO}(2)$:
$$
\pi_1(\theta) = \begin{bmatrix}\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \\
\pi_2(\theta) = \begin{bmatrix}\cos(2\theta) & -\sin(2\theta) \\ \sin(2\theta) & \cos(2\theta) \end{bmatrix} \\
$$
These are inequivalent irreducible representations (over the reals), so there are no non-trivial linear equivariant maps A. But are there non-linear maps $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that are equivariant? Is there a name for this type of thing? In what field (if any) are these maps studied? When (if ever) do they exist?
I'm guessing that if we assume that $G$ is an algebraic group, there can be polynomial equivariant maps, but I may be wrong.
 A: The ring of polynomial functions $V\to K$ is called the coordinate ring and is denoted $K[V]$ (although personally, I think it could've been denoted $K[V^*]$ instead, because polynomials are sums of products of functionals $V\to K$ no?) It is so-called because if $x_1,\cdots,x_n$ are coordinate projections $V\to K$ in some basis then $K[V]=K[x_1,\cdots,x_n]$.
If $V$ is a representation of $G$, then $G$ acts by automorphisms on the ring $K[V]$. As an algebra and as a representation, if $K$ has characteristic $0$ then the coordinate ring $K[V]$ is isomorphic to the full symmetric algebra ${\rm Sym}(V^*)$ which decomposes as an infinite product $\bigoplus_d {\rm Sym}^d(V^*)$.
The ring of polynomial functions $V\to W$ is then $K[V]\otimes W$. Then the $G$-equivariant polynomial functions $V\to W$ are the $G$-invariants of $K[V]\otimes W$, so it reduces to computing the invariants of the graded components $({\rm Sym}^d(V^*)\otimes W)^G$, which is a task for linear representation theory; if one has the character table of $V$ and $W$ (and $G$ is finite) then for any given $d$ one can reduce finding its dimension to a finite numerical calculation. Indeed, given ordered bases for $V$ and $W$ and explicit matrices for elements of $G$, one can compute a basis for the $G$-invariants of the $d$th component.
Maybe look into invariant theory to see if it covers more. Dunno about nonpolynomial stuff.
