Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results on the orbits inside $P$.

Actually, I would like to know which (say) permutation representations of the parabolic $P\to Sym(\Omega)$ admit an extension to a $P$-equivariant map of sets $G\to Sym(\Omega)$. But I do not know in which context this question falls.

I would already be very happy for an answer in the case $G=GL_3(\mathbb{F}_p)$ and the $(2,1)$- parabolic $P=(GL_2(\mathbb{F}_p)\times GL_2(\mathbb{F}_p))\ltimes \mathbb{F}_p^2$. I want to play around with a group-theoretic version of connection and parallel transport on the Grassmanian $Gr(3,2)$ over $\mathbb{F}_p$.

Thanks and greetings, Simon

Given a reductive group $G$ over a finite field and a parabolic subgroup $P$ , I wonder what are the orbits in $G$ under the adjoint action of $P$. This should be standard, but I can only find results on the orbits inside $P$.

Actually, I would like to know which (say) permutation representations of the parabolic $P\to \operatorname{Sym}(\Omega)$ admit an extension to a $P$-equivariant map of sets $G\to \operatorname{Sym}(\Omega)$. But I do not know in which context this question falls.

I would already be very happy for an answer in the case $G=\operatorname{GL}_3(\mathbb{F}_p)$ and the $(2,1)$- parabolic $P=(\operatorname{GL}_2(\mathbb{F}_p)\times \operatorname{GL}_2(\mathbb{F}_p))\ltimes \mathbb{F}_p^2$. I want to play around with a group-theoretic version of connection and parallel transport on the Grassmanian $\operatorname{Gr}(3,2)$ over $\mathbb{F}_p$.

Thanks and greetings, Simon