\documentclass{article}
 \usepackage{amsmath}
 \usepackage{amssymb}
%\input{newcommands}




\begin{document}
\newcommand{\op}{\operatorname}
\newcommand{\sira}{\stackrel{\sim}{\rightarrow}}
\newcommand{\DC}{\mathbb C}

\begin{center}
  {\Large Corrections for 
    W. Soergel: $\mathfrak n$-Cohomology of limits of discrete series \cite{So-NK},
   2024 edition}
\end{center}
In this note I want to point out two flaws 
in the argumentation of said article  and explain why they have
no effect on the main result. The first flaw was adressed
already in a the corrigenda \cite{So-NKK}. 
In addition, I want to add a correction to the example added in \cite{So-NKK}.
For making me aware of both additional corrections since the
corrigenda published in ERT I thank Jin Lee, currently at Duke. 

First, preceding Theorem 3.2
an equivalence of categories 
$S=S_\psi:\mathcal O_\psi\sira \mathcal O'_\psi$ is cited 
from \cite{So-N}. However, there is no such equivalence mapping
every Verma  to the same Verma. Rather the equivalence constructed there will
map $M(x\cdot\lambda)$ to $M(x^{-1}\cdot\lambda)$ for $\lambda$
the highest weight of a projective Verma alias the dominant weight of the
corresponding Weyl group orbit. 
This means that Proposition 3.9 breaks down and instead of
Theorem 3.2
we only get 
$${\op{H}}^i(\mathfrak n,T_\psi^\chi N)_\lambda\cong
\op{Ext}^i_{U_\psi}(T_\chi^\psi(M(\tilde \lambda))
\otimes_Z Z/\psi,N)$$
However I claim that 
instead of Proposition 4.1 we have, now with $\psi=Z^+$, an isomorphism
$$\Delta (T_\chi^\psi(M(\tilde \lambda))\otimes_Z Z/\psi)\cong j_!\mathcal A$$
and thus there is no effect of this error on the final result.
For this we need to alter the proof of
Proposition 4.1 
and show in the notation used there
rather that
$(T_\chi^\psi(M(\tilde \lambda))\otimes_Z Z/\psi)$ is a projective cover of
$M_y$ in the subcategory $\langle M_x\mid x\in D\rangle$ of $\mathcal O'_\psi$.
Well, for $N$ in this subcategory we find
$$
\begin{array}{lll}
\op{Hom}_{\mathfrak g}(T_\chi^\psi(M(\tilde \lambda))\otimes_Z Z/\psi, N)
&=&\op{Hom}_{\mathfrak g}(T_\chi^\psi(M(\tilde \lambda)), N)\\
&=&\op{Hom}_{\mathfrak g}(M(\tilde \lambda), T^\chi_\psi N)
\end{array}
$$
Since $T^\chi_\psi N$ has to belong to the subcategory $\langle
M(\lambda)\rangle$ 
of $\mathcal O'_\chi$ generated in the same way as above,
by definition of $M(\tilde \lambda)$ this is exact as a functor of $N$.
In addition it maps each $M_x$ for $x\in D$ to a one-dimensional space and
thus has to be the projective cover of $M_y$ as claimed.
Now it is clear how to rewrite the proof of Corollary 4.2, and from there
on one can continue as written in the article.

Second, we should on page three of \cite{So-NK} only ask $\mathcal V\subset \mathcal W$ 
to be a subset which together with any two elements $x<y$ with $l(x)+2=l(y)$ the
whole interval. This is the only thing we need to define the
complex $C^\bullet \mathcal V$ and this is the only thing we need to check for
$\mathcal V=\mathcal V(A,D,c)$ and in the case $A=A(Y)$.
Thanks to Jin Lee for pointing this out. 

Let me add a sample calculation.
Take the real form $\op{SU}(2,2)\subset \op{SL}(4;\DC)=G$ 
and take as a first Borel $B_{\op{std}}$ the upper triangular matrices.
Let $r,s,t$ be the standard generators of the Weyl group, so that
$r,t$ generate the compact Weyl group. The 
$K_\DC$-orbit $Y$ of $str$ in the flag variety 
$X$ is closed. If we now take instead the 
Borel $B$ fixing this point $strB_{\op{std}}$, then as the $B$-orbits 
    meeting $Y$ we find those with parameters $1$, $tst$, $rsr$ and $strs$. 
So this is our set $A=A(Y)$. Let us concentrate on $\chi$ the most singular 
central character, so $D=\mathcal W$. 
Now from our  $B$-orbits 
    meeting $Y$,
the first meets $Y$ in codimension zero and  the other three in codimension two.
So if $\mathfrak n$ is the nilradical of $\op{Lie}B$ and we compute
the $\mathfrak n$-cohomology of the
most degenerate limit of principal series
${\op{H}}^n(T^\chi;i_*\underline Y)$
for $\chi$ the most singular central character, 
we find nonvanishing  complexes $\mathcal V(A,D,c)$ 
only for $c=0$ and $c=2$. The following had to corrected from the published corrigenda, thanks for Jin Lee for pointing out the mistakes. 
For $c=0$ the complex has only one entry $\DC$ sitting in degree zero
and gives a one-dimensional
contribution to  
${\op{H}}^2$ (not ${\op{H}}^4$ as wrongly stated in \cite{So-NKK}).
For $c=2$ the complex has a one-dimensional 
 entry  in degree four, coming from to $strs$, and a 
two-dimensional 
 entry  in degree three, coming from to $rsr$ and $tst$, 
and the differential is not zero. This gives a one-dimensional
contribution to  ${\op{H}}^3$ (not ${\op{H}}^5$ as wrongly stated in \cite{So-NKK}). We find no more cohomology
than that.



%\bibliographystyle{amsalpha}\bibliography{pub}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{Soe97}

\bibitem[Soe89]{So-N}
Wolfgang Soergel, \emph{$\frak n$-cohomology of simple highest weight modules
  on walls and purity}, Inventiones \textbf{98} (1989), 565--580.

\bibitem[Soe97]{So-NK}
\bysame, \emph{On the $\mathfrak n$-cohomology of limits of discrete series
  representations}, Representation Theory (An electronic Journal of the AMS)
\textbf{1} (1997), 69--82.

\bibitem[Soe15]{So-NKK}
\bysame, \emph{Corrections to: On the $\mathfrak n$-cohomology of limits of discrete series
  representations}, Representation Theory (An electronic Journal of the AMS)
  \textbf{19} (2015), 1--2.
\end{thebibliography}
% scp /home/wolfgang/Dokumente/Skripten/Skripten/KorrLDS24.pdf soergel@tux00:/webserver/home/soergel/PReprints/KorrLDS24.pdf
\end{document}