Let us take $G=\operatorname{Gl}(n,\mathbb{C})$ (considered as a linear algebraic group). Let us take $x \in G$: we know that its orbit $\mathcal{O}_x$ under the conjugation action is isomorphic (as algebraic varieties) to $$ G/C_G(x) \cong \mathcal{O}_x $$ via the action map where $C_G(x)$ is the centralizer group. In particular the action map realizes $\mathcal{O}_x$ as the base space of a principal $C_G(x)$ bundle.
I was trying to determine what happens if we take the following more general situation: we take $$n_1+ \dotsb+ n_k=n $$ and we fix partitions $\lambda_1,\dotsc, \lambda_k$ of $n_1,\dotsc, n_k$ respectively. To each such partition we can associate in a unique way an (uppertriangular) unipotent Jordan matrix, in the canonical way, which we denote $J_{\lambda_i}$.
We put $n_1 \geq \dotsb \geq n_k$.
We then call $U_{\lambda_1,\dotsc, \lambda_k}$ the set of elements of $\operatorname{Gl}(n,\mathbb{C})$ with $k$ different eigenvalues such that the Jordan form of the $i$th eigenvalue is given by $J_{\lambda_i}$.
If we denote $$W=\{ (x_i) \in \mathbb{G}^k_m \mathrel\vert x_i \neq x_j \ ,\ i \neq j\} $$ we have a map $$W \times Gl(n,\mathbb{C}) \to U_{\lambda_1,\dotsc, \lambda_k} $$ which sends the element $((x_i),g) $ to $g $ applied to the block upper triangular matrix with blocks given by $x_iJ_{\lambda_i}$.
The morphism is surjective and so $ U_{\lambda_1,\dots \lambda_k}$ is constructible: I've tried to prove some of its geometrical properties but I didn't get much. This morphism in general does not seem to be a $C_G(x) $ principal bundle to me: as the fiber can happen to be bigger than elements of the form $$((x_1, \dotsc, x_k),h) $$ with $h$ in the associated centralizer. Is there a way to change the approach and make this variety a base of a principal $C_G(x)$ bundle?
I was not even able to prove whether this variety is smooth or not (apart from the regular semisimple case $\lambda_i=1$ for example or other degenerate ones). Is there a reference in the literature for this?
EDIT: The question I had in mind more precisely was this. If we take a single orbit, we have that the map $G \to \mathcal{O}_x$ given by the action on $x$ is a principal $C_G(x)$ bundle.
This implies locally on $\mathcal{O}_x$ (even Zariski locally as centraliser in general linear groups are special) that we have section. For each $z \in \mathcal{O}_x$ we have $z \in V$ open with $\psi_V:V \to G$ such that $\psi(v) \cdot x=v$ with $v \in V$.
Is there a similar morphism for these Lusztig strata?Can we find locally on $U_{\lambda_1, \dots \lambda_k}$ morphisms such as the $\psi_V$? At least to me the biggest problem seems to replace the role of the $x$ in the orbit case with suitable representatives of coniugacy classes in an open $V$.
