# About an application of BBD decomposition theorem

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.*

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?)

Because $$\pi$$ is smooth, the $$Y_i$$s in the decomposition theorem must all be the entire space $$G/Q$$.
Because $$G/Q$$ is simply-connected, the local systems are all constant.
The fiber of $$G/B \to G/Q$$ at the identity is $$Q/B$$, and this is known to be a sum over $$W^Q$$ by the standard cell decomposition.