In the study of certain moduli spaces of $p$-divisible groups I came across the following twisted version of a Springer fiber, and I was wondering whether some expert on algebraic groups/algebraic geometry would have an idea whether this has been studied before (smooth locus, irreducibility, singularities). I am not asking for helping me to study this scheme but references would be very appreciated (if they exist).

Let $V$ be a finite-dimensional vector space over a finite field $k = {\mathbb F}_q$ and consider $G = GL(V)$ as an algebraic group over $k$. Let $F\colon G \to G$ be the Frobenius over $k$. Let $P$ be a parabolic subgroup of $G$ (defined over $k$), ${\mathfrak p} \subset {\rm End}(V)$ its Lie algebra, and let $x \in {\mathfrak p}$ be nilpotent. I am asking about the following closed subscheme of $G/P$:

$X$ = {$gP \in G/P \mid F(g)^{-1}xg \in {\mathfrak p}$}

I hope it is clear how $X$ is defined in terms of $R$-valued points for an arbitrary $k$-algebra $R$ (and therefore what its scheme structure is).

**Remark**: Replacing $F$ by ${\rm id}$ would make $X$ into a Springer fiber, hence the title. But this might be misleading because twisted conjugation is not an endomorphism of the Lie algebra ${\rm End}(V)$ and my definition of $X$ uses that there is a canonical embedding of $G$ into its Lie algebra (and of $P$ into ${\mathfrak p}$). This is of course very special for $G = GL(V)$.

A better way to understand $X$ (and to generalize it to arbitrary reductive groups) might be the following: Let $G$ be a reductive group over a finite field $k$ and let $V$ be a rational representation of $G$ over $k$, considered as $G$-*right* module. Let $V^*$ its dual representation and consider $V \otimes V^*$ as a representation of $G \times G$. Embedding $G$ into $G \times G$ by $g \mapsto (F(g),g)$ we obtain an representation $\rho$ of $G$ on $V \otimes V^*$ by restriction. Let $U \subset V$ be a subspace, let $P \subset G$ be the stabilizer of $U$. Set $W = U \otimes V^* + V \otimes U^0$, where $U^0 \subset V^*$ is the orthogonal of $U$, and let $x \in W$. Then

$X = \{gP \in G/P \mid x\rho(g) \in W\}$

is a generalization of the above variety (at least if $P$ is maximal parabolic).