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Fix $G$ a finite dimensional reductive group and $\lambda$ a weight. Apparently the category of $\mathcal{D}_\lambda$ modules on $G/B$ is equivalent to the category of $\mathcal{D}_0$ modules on $G'/B'$ for a different group $G'$, whose Weyl group $W'$ consists of the elements $w\in W$ with $w\lambda-\lambda$ integral. $G'$ is something like the Langlands dual of the centraliser of $\lambda$ inside $G$.

Apparently this is due to Lusztig; in his orange book it is covered in the language of monodromic sheaves.

Question: is there a modern reference for this?

Edit to clarify, in light of the comments: I think it should be true for any $\lambda\in\mathfrak{g}^*$, and $w\lambda-\lambda$ integral means that $\langle w\lambda-\lambda,\alpha^\vee\rangle\in\mathbf{Z}$ for all coroots $\alpha^\vee$. However, I might be wrong and there are are in fact more conditions.

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    $\begingroup$ An integral weight is usually one that belongs to the $\mathbb{Z}$-span of the fundamental weights. If $\lambda$ itself is non-integral, then there might not be any non-identity $w\in W$ for which $w\lambda-\lambda$ is integral (think of $\lambda$ with a very small but nonzero length). $\endgroup$
    – Sam Hopkins
    Commented Nov 23, 2020 at 16:25
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    $\begingroup$ Right: often "weight" is used synonymously with "integral weight"; but here I think it should mean the more inclusive thing (corresponding to infinite-dimensional representations). $\endgroup$
    – Sam Hopkins
    Commented Nov 23, 2020 at 16:53
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    $\begingroup$ I think your edit is almost right but you want that $w\lambda-\lambda$ pairs integrally with coroots $\alpha^\vee$ not roots. $\endgroup$
    – Sam Hopkins
    Commented Nov 23, 2020 at 17:27
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    $\begingroup$ In that case, I think you want something like the parabolic-singular Koszul duality of Beilinson-Ginzburg-Soergel, or rather a combination of that with the Koszul duality as formulated by Soergel ("Koszul duality and Langlands' philosophy", cf Bezrukavnikov-Yun) relating flag varieties for Langlands dual groups. You might also look for something closely related at arxiv.org/abs/1904.01176 $\endgroup$
    – David Ben-Zvi
    Commented Nov 23, 2020 at 20:26
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    $\begingroup$ You are right that this is in Lusztig's book, and this is the "modern" reference AFAIK. One thing that is not as well-known as it should be is that in the case of category O (say $N$-equivariance) this follows from Soergel's 1990 J|AMS paper in a very beautiful way. This is not mentioned in the paper but is commented on in the MathSciNet review. $\endgroup$
    – Geordie Williamson
    Commented Nov 23, 2020 at 20:29

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