**Context:** Let $B_n$ be the space of symmetric bilinear forms on $\mathbb{R}^n$ and $L_n\subset B_n$ be the subset of non-degenerate forms of Lorentzian signature $(-,+,\ldots,+)$. Let $T$ be a finite dimensional real vector space. Both $B_n$ and $T$ carry linear representations of $GL^+(n,\mathbb{R})$, where $+$ means the connected component of the identity, with the action on $B_n$ restricting to $L_n$. We assume that, for $g\in GL^+(n,\mathbb{R})$, the corresponding linear maps acting on $B_n$ and $T$ are polynomial in the components of $g$ up to a multiple of $\left|\det g\right|$ to a rational power. For $h\in L_n$, consider its components $h_{ab}$ (using some fixed basis on $\mathbb{R}^n$, say) as functions on $L_n$. I can also define another set of functions which are the components $\varepsilon_{a_1\cdots a_n}$ of the Levi-Civita tensor, defined as a solution to the equation $$\varepsilon_{a_1\cdots a_n} \varepsilon_{b_1\cdots b_n} = \text{(antisymmetrization over $a_1\cdots a_n$ of)} ~ h_{a_1 b_1} \cdots h_{a_n b_n}.$$

Question:If I have a (non-linear)smoothequivariant map $A\colon L_n \to T$, can I conclude that $A$ is actuallypolynomialin the functions $h_{ab}$ and $\varepsilon_{a_1\cdots a_n}$ (up to a multiple of a rational power of $\left|\det h_{ab}\right|$) on $L_n$?

I believe that the answer is Yes. Unfortunately, I don't know which (probably well-known) result implies that. In the absence of a direct answer, some hints about how to find one in the vast literature on Invariant Theory would also be appreciated.

**Note:** Most of the literature in Invariant Theory seems to already make the assumption that all maps involved are polynomial, rational or algebraic. On the other hand, my question is about reducing to one of the contexts starting with smoothness and some other assumptions.