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\begin{document}
For your diagram, I think that's essentially it, but it will not
commute without taking additional care with the maps.
More precisely,
we will have a commutative diagram
$$
\begin{array}{ccc}
\op{Der}(\pi^*\pi_!\mathcal F,\mathcal G)&\sira&
\op{Hom}_C(C\otimes_{C^s} M,N)\\
\da\wr&&\da\wr\\
\op{Der}(\pi_!\mathcal F,\pi_* \mathcal G)&\sira&
\op{Hom}_{C^s}(M,N)\\
\da\wr&&\da\wr\\
\op{Der}(\mathcal F,\pi^!\pi_* \mathcal G)
&\sira&
\op{Hom}_C(M,\op{Hom}_{C^s}(C,N))
\end{array}
$$
for $M,N$ the hypercohomologies,
and following this through and using
that $C$ is a free $C^s$-module with basis $1$ and $\alpha$,
we will see that does on the right 
vertical something a little bit tricky.
\end{document}
\begin{center}
  {\Large Corrections for 
 W. Soergel: On the relation between intersection cohomology and representation theory 
in positive characteristic, pp. 311-335 in Journal of Pure and Applied Algebra
152 (2000) }
\end{center}

\begin{enumerate}
\item The argument on page 329 at the bottom needs fixing.
Thanks are due to Masaharu Kaneda for pointing this out
and insisting. 
Let us reprove Proposition 3.2.6.
First we treat the case that $\mathcal F = i_{n!} \underline{B/B}$ is
the skyscraper at the one-point cell.
In this case the claim follows from the fact that $i_{n!} i_n^! \mathcal E \rightarrow
\mathcal E$ induces an injection on hypercohomology, which follows from the degeneration of
the spectral sequence computing $\mathbb H^\bullet \mathcal E$
explained in the proof of 3.2.6.
Then, for the general case, it will be sufficient to check the commutativity of the
following diagrams, for $\mathcal A \in \mathcal D (G/P_s)$ and $\mathcal F \in \mathcal D (G/B):$
\begin{displaymath}
 \xymatrix{
\op{Hom}_{\mathcal D} (\pi^\ast \mathcal A, \mathcal F)\ar[d]^-{\wr} \ar[r]& \op{Hom} (\mathbb H^\bullet
\pi^\ast \mathcal A, \mathbb H^\bullet \mathcal F)\ar[d]\\
\op{Hom}_{\mathcal D} (\mathcal A, \pi_\ast \mathcal F) \ar[r] & \op{Hom} (\mathbb H^\bullet \mathcal A,
\mathbb H^\bullet \pi_\ast \mathcal F)
}
\end{displaymath}
for the map on the right
 coming from $\mathcal A \rightarrow \pi_\ast\pi^\ast \mathcal A$,
and
\begin{displaymath}
 \xymatrix{
\op{Hom}_{\mathcal D} (\mathcal F, \pi^!\mathcal A) \ar[r]\ar[d]^-{\wr} & \op{Hom}_C (\mathbb H^\bullet
\mathcal F, \mathbb H^\bullet \pi^! \mathcal A) \ar[d]^-{\wr}\\
\op{Hom}_{\mathcal D} (\pi_! \mathcal F, \mathcal A) \ar[r] & \op{Hom}_{C^s} (\mathbb H^\bullet \pi_! \mathcal F,
\mathbb H^\bullet \mathcal A)
}
\end{displaymath}
Here the point is to construct dually to
Proposition 4.1.1 a canonical isomorphism
$\op{Hom}_{C^s}(C,\mathbb H^\bullet \mathcal A)\sira \mathbb H^\bullet\pi^!
\mathcal A$ and show that the resulting diagram will commute.
With these diagrams, a non-injective case in 3.2.6 would lead
 to a noninjective case with
$\mathcal F$ the skyscraper, which we have already shown to be impossible.
\item
 I add some details concerning the proof
of  Corollary 1.0.3. The point is that if Lusztig's conjecture is
 ok, then the projectives of the modular $\mathcal O$ 
have the same Verma flag multiplicities as 
they do for the classical non-modular $\mathcal O$,
given by KL-polynomials for the finite Weyl group.
However, if the decomposition theorem would not be ok
at some step of the inductive construction of these 
special indecomposable complexes now called parity sheaves, 
and we take the first such step, 
these projectives
 would turn out bigger for the very parameter this step   of their 
 inductive construction leads to.
\end{enumerate}

% scp KorrICP.pdf soergel@tux00:/webserver/home/soergel/PReprints/KorrICP.pdf
\end{document}