There is a following proposition in the famous paper on Koszul duality by Beilinson, Ginzburg and Soergel:

*let $G$ denote a semisimple complex Lie group, let $B$, $Q$ and $W^Q$ denote a pair of a Borel and parabolic subgroups and the corresponding parabolic Weyl group respectively, let $pi$ denote an evident morphism $G/B to G/Q$. For a variety $X$ let $mathbb C_X$ denote a constant sheaf of 1-dimensional vector spaces on X. Then the derived pushforward $pi_* mathbb C_{G/B}$ is $$bigoplus_{x in W^Q} mathbb C_{G/B}(-2l(x)).$$*

Authors claim that it’s a consequence of BBD decomposition theorem which is known to me in the following formulation:

*for a proper morphism $f: X to Y$ of algebraic varieties there is an isomorphism

$$R f_{*}left(mathrm{IC}_{X} cdotright) simeq bigoplus_{k}^{text {finite }} i_{k *} mathrm{IC}_{Y_{k}}left(L_{k}right)^{cdot}left(l_{k}right),$$

where $Y_k$, $L_k$ and $l_k$ are some locally closed subvarieties, local systems on them and integer numbers respectfully.*

(See also here for the stratified version.)

Can someone explain me how to derive BGS’s claim about $G/B$ and $G/Q$ from this theorem?

(Using the obvious stratification (by B-orbits) one can see that it is sufficient to prove the desired result for a big cell $F$ in G/Q (i.e. to prove the claim that $Rf_*(mathbb C_{G/B})|_F = bigoplus_{x in W^Q} mathbb C_F(-2l(x))$), isn’t it?)