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In 1976, Luna proved the following important theorem of smooth invariant theory:

Let $G$ be a real reductive Lie group and a representation of $G$ on a real finite dimensional vector space $V$. Also, let polynomials $(p_i)_{i=1}^n$ be a generating set for the $G$-invariant polynomials on $V$.

Then, any $G$-invariant smooth function $f \in C^\infty(V)^G$ that is also constant on the joint level sets of the $p_i$ (that is, $f$ cannot separate any points of $V$ that are not already separated by invariant polynomials) is necessarily of the form $f = F(p_1,\ldots,p_n)$, where $F$ is a smooth function of $n$ variables.

  • Luna, D. Fonctions différentiables invariantes sous l'opération d'un groupe réductif. Annales de l'institut Fourier 26, 33-49 (1976) http://dx.doi.org/10.5802/aif.599

Let me call the subset of the smooth invariants that satisfy the hypotheses of Luna's theorem stable, and denote them by $C^\infty(V)^G_{\mathrm{stab}} \subset C^\infty(V)^G$. This is consistent with term stable orbit referring to a joint level set of the invariant polynomials $p_i$, as I was informed in an answer to my previous question. Each stable orbit may contain a finite number of distinct actual $G$-orbits.

So, Luna's theorem gives a complete description of smooth stable invariants $C^\infty(V)^G_{\mathrm{stab}}$, but not of all smooth invariants $C^\infty(V)^G$. My question is the following:

Q: What progress has there been in explicitly describing the structure of all smooth invariants $C^\infty(V)^G$?

My guess is that algebraically $C^\infty(V)^G$ should be some kind of finite extension of $C^\infty(V)^G_{\mathrm{stab}}$, which one should be able to describe using some representation theoretic data related to $G$ and $V$. A good partial answer might be a guess at this representation theoretic data! But of course, this simple idea is complicated by the need to keep everything smooth.

Consider the fundamental vector representation of the orthochronous Lorentz group $G = SO^\uparrow(1,q) \subset SO(1,q)$ acting on $V = \mathbb{R}^{1+q}$. In the language of special relativity, orthochronous Lorentz transformations are distinguished by their inability to exchange future- and past-directed timelike vectors. There is only one independent polynomial $G$-invariant for $v=(v_0,v_1,\ldots,v_q) \in V$, namely $\langle v, v \rangle = -v_0^2 + v_1^2 + \cdots + v_q^2$. For $\langle v, v \rangle \ge 0$, stable orbits coincide with actual orbits, while for $\langle v, v \rangle < 0$, a stable orbit splits into two orbits, distinguished by the sign of $v_0$. So an arbitrary smooth $G$-invariant function $f \in C^\infty(V)^G$ has the form $$ f(v) = F(\langle v, v\rangle) + \Theta(+v_0) H_+(\langle v, v\rangle) + \Theta(-v_0) H_-(\langle v, v\rangle) , $$ where $F(x)$ is an arbitrary smooth function, $\Theta(t)$ is the Heaviside step function and $H_\pm(x)$ are arbitrary smooth functions that vanish identically for $x>0$ (in particular they are flat at $x=0$). They are non-unique due to the obvious relation $\Theta(+v_0) + \Theta(-v_0) = 1$. In a sense, $\Theta_\pm = \Theta(\pm v_0)$ do generate all smooth invariants starting from smooth stable invariants, but it's not true that $C^\infty(V)^G$ is isomorphic to the extension $C^\infty(V)^G_{\mathrm{stab}}[\Theta_+,\Theta_-]/(\Theta_+ + \Theta_- = 1)$.

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  • $\begingroup$ Now that I think about it it is not a good idea to call the real points of a complex orbit a "stable orbit" since this clashes with the notion of stability in GIT. On the other hand, there are stable conjugacy classes which are used in this sense. $\endgroup$ Aug 28, 2017 at 15:33

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