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?)


One approach:

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.

To calculate the multiplicities in each degree, it therefore suffices to evaluate the stalk at any one point, i.e. to calculate the calculate the cohomology of any fiber.

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.


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