Now i am a little confused, so you say the statement holds also for the full center?

I edited my comment above and replaced Zariski by etale.

Ok, i am sorry but maybe my question was to naive. First i really thought about the obvious (naive) definition of a fiber bundle in the Zariski topology, of course if there are nice results in the form of: We have an algebraic morphism $E\to B$ which is a fiber bundle in the etale topology. If $F$ and $B$ are projective then is $ E$ (for maybe a very restrictive class of varieties $B$ and $F$) then so is $E$ I would be glad to hear about it. @ZhuangXiaobo thank you for the reference! – O. Straser 3 hours ago

I would say locally trivial in the Zariski topology!

Yes, at least roughly: Soergel conjectures that the category of smooth admissible representations of a real reductive group $G$ can be described by the equivariant derived category of the so-called Adams-Barbasch-Vogan parameter space (equivariant with respect to some action of the Langlands-dual group of the complexification of $G$) Ok, this has nothing to do with geometric Langlands on first sight, but he also mentions that similar conjectures should hold in the $l$-adic case.

You are absolutely right, i meant $k<n$. Thank you very much for your both corrections!