Automorphism groups of multilinear forms and maps (Q584348)

Let \(\{\theta_ i: V\to k\}\) be a set of multilinear forms on a vector space V over an algebraically closed field \(k,\{\mu_ i: V\to V\}\) be a set of multilinear maps from V to V. The subspae U of V is called s- isotropic with respect to \(\{\theta_ i\}\) if \(\theta_ i(v_ 1,...,v_ r)=0\) when s of the \(v_ j\) lie in U. The subspace \(U\subseteq V\) is called an inner s-ideal if \(\mu_ i(v_ 1,...,v_ r)\subseteq U\) when s of \(v_ j\) lie in U, and \(=0\) when \(s+1\) of \(v_ j\) lie in U. Theorem 1: If \(\{\theta_ i\}\) are of degree \(r\geq 3\) then either \(Aut_ k\{\theta_ i\}\) is finite or V contains a nonzero s-isotropic subspace. Theorem 2: If \(\{\mu_ i\}\) are of degree \(r\geq 3\) then either \(Aut_ k\{\mu_ i\}\) is finite or V contains a nonzero inner s-ideal. Question: Let \(G=G(k)\) be a simple algebraic group, V its kG-module of lowest dimension, \(\theta\) a multilinear form on V for which \(Aut_ k \theta =G\). Is it true that stabilizers of flags of isotropic spaces are exactly parabolic subgroups of G(k)?