Comparing cohomology of quotient by algebraic group and Borel subgroup

Question

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.

Answer 1

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.

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).