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

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

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?)