Let $G$ and $\tilde G$ be connected complex reductive Lie groups, $G$ semisimple. Let $Λ^+$ be the monoid of dominant weights for a positive root system $Δ^+$, and let $l(w)$ be the length of a Weyl group element $w$. Let $V_λ$ denote an irreducible $G$-module of highest weight $λ\inΛ^+$. For any closed embedding $ι:\tilde G\subset G$, we consider Property (A): $\quad\forallλ\inΛ^+,\exists q\in\mathbb{N}$ such that $V_{qλ}^{\tilde G}\ne0$. A necessary condition for (A) is for $G$ to have no simple factors to which $G$ projects surjectively. We show that this condition is sufficient if $\tilde G$ is of type ${\bf A}_1$ or ${\bf E}_8$. We define and study an integral invariant of a root system, $\ell_G=\min\{\ell^λ:λ\inΛ^+\setminus\{0\}\}$, where $\ell^λ=\min\{l(w):wλ\notin{\rm Cone}(Δ^+)\}$. We derive the following sufficient condition for (A), independent of $ι$: $$ \ell_G - \#\tildeΔ^+ > 0 \;\Longrightarrow\; (A). $$ We compute $\ell_G$ and related data for all simple $G$, except ${\bf E}_8$, where we obtain lower and upper bounds. We consider a stronger property (A-$k$) defined in terms of Geometric Invariant Theory, related to extreme values of codimensions of unstable loci, and derive a sufficient condition in the form $\ell_G - \#\tildeΔ^+ > k$. The invariant $\ell_G$ proves too week to handle $G=SL_n$ and we employ a companion $\ell_G^{\rm sd}$ to infer (A-$k$) for a larger class of subgroups. We derive corollaries on Mori-theoretic properties of GIT-quotients.