Let me first give some definitions for reference.
Let $\mathbf G$ be a connected reductive group over an algebraic closure $\overline{\mathbb F}$ of a finite field $\mathbb F_p$, where $p$ is a prime number. Let $q$ be a power of $p$, and assume that $\mathbf G$ comes together with an $\mathbb F_q$-structure, given by a Frobenius morphism $F$. We choose a pair $(\mathbf T, \mathbf B)$ consiting of an $F$-stable maximal torus and an $F$-stable Borel subgroup containing it. Denote by $W$ and $S$ the subsequent Weyl group and set of simple reflections. Note that the Frobenius $F$ acts on $W$ and preserves $S$. For $I$ a subset of $S$, let $W_I$ denote the subgroup of $W$ generated by $I$, and let $\mathbf P_I := \mathbf B W_I \mathbf B$ be the associated standard parabolic subgroup of type $I$.
A parabolic subgroup $\mathbf P$ of $\mathbf G$ is said to be of type $I$ if it is conjugated to $\mathbf P_I$. The variety classifying such parabolic subgroups is denoted by $\mathrm{Par}_I$. The group $\mathbf G$ acts on the product $\mathrm{Par}_I\times \mathrm{Par}_{F(I)}$ by conjugation. For $w\in W$, we denote by $\mathcal O_I(w)$ the orbit of $(\mathbf P_I, \,^w\mathbf P_{F(I)})$ under this action, it only depends on the double coset $W_IwW_{F(I)}$. We finally define the associated Deligne-Lusztig variety by the formula
$$X_I(w) = \{\mathbf P\in \mathrm{Par}_I \,|\, (\mathbf P,F(\mathbf P))\in \mathcal O_I(w)\}$$
I have seen some examples of this construction in a few cases. In a French lecture note on the matter, I saw that the Deligne-Lusztig varieties for $\mathrm{GL_2}$ and $\mathrm{SL_2}$ with a split $\mathbb F_q$-structure are the same, namely we have $$X(\mathrm{id}) \simeq \mathbb P_1(\mathbb F_q) \quad \quad X(s)\simeq \mathbb P_1 \setminus \mathbb P_1(\mathbb F_q)$$
It made me wonder, can the same be said of Deligne-Lusztig varieties for $\mathrm{GL}_n$ and $\mathrm{SL}_n$ for a general $n$ ? Is it also true in the case of a non-split $\mathbb F_q$-structure (ie. for $\mathrm{U_n}$ and $\mathrm{SU}_n$) ?
If so, is it true that the variety $X_I(w)$ only depends on the root system associated to $\mathbf G$ together with the Frobenius action on it ?
NB: I am aware that we may also define another variety $Y_I(w)$ over $X_I(w)$, by considering cosets $g\mathbf U_I$ instead of $g\mathbf P_I$, where $\mathbf U_I$ is the unipotent radical of $\mathbf P_I$. In this case, the varieties $Y_I(w)$ do depend on $\mathbf G$, as we obtain the Drinfeld curve for $\mathrm{SL}_2$ and something a little bigger in the case of $\mathrm{GL}_2$ with split structure.
