The Hilbert Mumford Criterion as in Wallach Theorem 3.24 says:
Let $G$ be a linearly reductive subgroup of $GL(n, \mathbb{C})$. Let $(\sigma, V)$ be a regular representation of $G$.
For a vector $v \in V$ and $Gw$ the closed orbit in $\overline{Gv}$. Then there exists an algebraic group homomorphism $\phi: \mathbb{C}^* \to G$ such that $\lim _{ z \to 0} \sigma (\phi(z))(v) \in Gw$.
Is some version of this true over the reals, or do we need the field to be algebraically closed?
The theorem holds over any perfect field $k$ and for any affine $G$-variety $X$ defined over $k$. More precisely, it states: Let $x\in X(k)$ be a $k$-rational point and $Y\subseteq\overline{Gx}$ be a closed $G$-invariant subvariety. Then there is a $1$-parameter $k$-subgroup $\lambda:\mathbb G_m\to G$ such that $\lim_{t\to0}\lambda(t)x\in Y(k)$.
This was proved independently by Kempf (Instability in invariant theory. Ann. of Math. (2) 108 (1978) 299–316, see Cor. 4.3) and Rousseau (Immeubles sphériques et théorie des invariants. C. R. Acad. Sci. Paris Sér. A-B 286 (1978) A247–A250).
The proof uses that over the algebraic closure of $k$ there is a $1$-parameter subgroup as in the Hilbert-Mumford criterion which is \emph{optimal} in a certain sense and therefore basically unique. Thus, a Galois theory argument yields one which is defined over $k$. The existence of a canonical $1$-PSG was conjectured already by Mumford.
