# ag.algebraic geometry – 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

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

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

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?)