Let $G$ be a finite dimensional semisimple algebraic group, and for $s\in W$ write $i_s: F_s=B^+sB^-/B^-\to F$ for the $s$th Bruhat cell in the flag variety $F$.

Then (in *Affine Kac-Moody Algebras and Semi-Infinite Flag Manifolds* bottom of p.165), Feigin-Frenkel claim that the local cohomology
$$H^*_{F_s}(F,\mathcal{O})\ =\ M^{s}_{s\star 0}\ =\ M^s_{s\rho-\rho}$$
is an $s$-twisted Verma module (ignoring shifts).

However, I'm pretty sure that local cohomology is the cohomology of $i_{s*}i_s^!\mathcal{O}$, which up to a shift is $i_{s*}\mathcal{O}$, which is known to give *dual* Vermas. e.g. see [arXiv:1412.0174][1] or Hotta et al.

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Is this a typo in Feigin-Frenkel? If it is, what is the correct formulation of their construction of dual Vermas? Their general statement is for $\lambda$ integral dominant, $H^*_{F_s}(F,\mathcal{L}(w\star \lambda))=M^s_{(ws)\star\lambda}$ up to shifts.


  [1]: https://arxiv.org/pdf/1412.0174.pdf