Let $X$ a variety over an algebraically closed field $k$ (which we can assume to be actually $k=\mathbb{C}$) and $G$ a connected reductive algebraic group acting freely on $X$ (we can actually assume $G=Gl\_n$). Can we relate somehow the (compactly supported) etale cohomology $H\_c(X/G,\overline{\mathbb{Q}}\_{\ell})$ and $H\_c(X/B,\overline{\mathbb{Q}}\_{\ell})$? Better


We have the morphism $f:X/B \to X/G$ which should be a fibration with fibre $G/B$ so the better possibile should be something like $$H\_c(X/B,\overline{\mathbb{Q}}\_{\ell})=H\_c(X/G,\overline{\mathbb{Q}}\_{\ell}) \otimes H\_c(G/B,\overline{\mathbb{Q}}\_{\ell}) $$


To get this or something similar my idea would have been to study the local systems $R^qf\_\*\overline{\mathbb{Q}\_{\ell}}$ and use the spectral sequence: $$E^{p,q}\_2=H^p\_c(X/G,R^qf\_\*\overline{\mathbb{Q}\_{\ell}}) \Rightarrow H^{p+q}\_c(X/B,R^qf\_\*\overline{\mathbb{Q}\_{\ell}}) .$$ However it seems to me that there should be no reason a priori for these local systems to be trivial or the sequence to abrupt at the second page.



Comparing cohomology of quotient by algebraic group and Borel subgroup
The spectral sequence degenerates on the second page since $X/B \to X/G$ is a smooth projective morphism (as $G/B$ is smooth projective) by a result of [Deligne and Blanchard](https://en.wikipedia.org/wiki/Leray_spectral_sequence#Degeneration_theorem).


The local systems are trivial because, first, they can be trivialized on open sets $U$ where the bundle is trivial, and then on the intersection between two open sets $U \cap V$, the gluing data is given by the action of some function $U \cap V \to G$ on $H^\* ( G/B, \overline{\mathbb Q}\_\ell)$, but this action is trivial since $G$ is connected so the gluing data is trivial.


So there is indeed a tensor product isomorphism as you suggest (though not usually one compatible with cup product structure).


