Leray-Serre spectral sequence for algebraic groups

Question

Let $G$ be a semisimple, simply-connected, complex algebraic group. Fix a Borel subgroup $B$ and let $P$ be a parabolic subgroup properly containing $B$. If $M$ is a $B$-module, then we have the Leray-Serre spectral sequence corresponding to the fibration $P/B\to G/B\to G/P$

$E^{p,q}_2=H^p(G/P, H^q(P/B,M))\implies H^{p+q}(G/B,M)$

The notation $H^•(G/B, M)$ means the sheaf cohomology of the sections of the bundle $G\times_B M\to G/B$ and similarly with $G,B,M$ replaced by another group $G$ with closed subgroup $B$ and $B$-module $M$.

My question is, how can the module on the right be seen as a graded version of the module on the left? What is the filtration?

Answer 1

In the vein of @MarkGrant's comment: Since this is a first quadrant spectral sequence, for a given pair $(p,q)$ the individual terms $E_2^{p,q},E_3^{p,q},E_4^{p,q},\ldots$ occurring on each subsequent page of the spectral sequence will eventually reach a stable value $E_\infty^{p,q}$. (The larger $p+q$ is, the more pages of the spectral sequence it may take for $E_r^{p,q}$ to reach its stable value.) In general $E_\infty^{p,q}$ will only be a subquotient of $E_2^{p,q}$. Then the statement that the spectral sequence converges to $H^*(G/B,M)$ means that $H^{p+q}(G/B,M)$ admits a decreasing filtration $F^* H^{p+q}(G/B,M)$ with $$F^p H^{p+q}(G/B,M) / F^{p+1} H^{p+q}(G/B,M) = E_\infty^{p,q}.$$ If you happen to know that the spectral sequence collapses at the $E_2$-page, and hence for all $p,q$ that $E_2^{p,q} = E_\infty^{p,q}$, then this tells you how to think of the $H^p(G/P,H^q(P/B,M))$ as the filtration layers of $H^{p+q}(G/B,M)$.