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Let $\pi$ denote a saturated set of weights. Let $S_q(\pi)$ denote the associated generalised $q$-Schur algebra. I was wondering if the following claim is true:

Claim: The algebra $S_q(\pi)$ is Koszul if and only if the decomposition numbers of $S_q(\pi)$ are given by the associated Kazhdan-Lusztig polynomials.

In characteristic zero and `sufficiently large primes' I believe this is true by work of Andersen-Jantzen-Soergel and Riche. But is it true or believed to be true over fields of arbitrary characteristic?

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    $\begingroup$ Kazhdan-Lusztig and Lusztig type conjectures are equivalent to a certain graded ring being positively graded and semi-simple in degree zero. Of course this is step 0 towards being Koszul! $\endgroup$ – Geordie Williamson Apr 14 '16 at 10:14
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    $\begingroup$ Perhaps the intro of "Modular Koszul duality" arxiv.org/pdf/1209.3760v1.pdf is helpful. In particular the remarks after Theorem 1.2.1 $\endgroup$ – Geordie Williamson Apr 14 '16 at 10:24
  • $\begingroup$ Thanks, Geordie! So I guess that means that (Koszul -->> KL polynomials) and that the converse is not necessarily true - but perhaps is expected to be true? $\endgroup$ – Chris Bowman Apr 14 '16 at 13:04
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    $\begingroup$ Basically yes. My understanding is that "positive grading with semi-simple degree 0" is the really hard part (i.e. the bridge the KL polynomials). I expect that in the examples one has once one knows Lusztig style conjectures then Koszulity should hold. (By the numerical criterion for example, as Ben mentions.) $\endgroup$ – Geordie Williamson Apr 14 '16 at 14:33
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Since Geordie already started the shameless self-promotion: I believe you can apply the results of section 1 of Canonical bases and higher representation theory to show that this will happen whenever the graded version of this $q$-Schur algebra (which you may need to believe a conjecture of Riche and Williamson to define) is Morita equivalent to a positively graded algebra. So asking for Koszulity is, as Geordie points out, actually overkill.

That said, I think Koszulity will likely follow in this case: an argument like in Notes on parameters of quiver Hecke algebras by Kashiwara should show that this positivity will only happen if the base change from characteristic 0 to characteristic $p$ sends simples to simples, in which case the numerical criterion shows that Koszulity is preserved.

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  • $\begingroup$ Sorry Ben, when you say "the graded version of this $q$-Schur algebra (which you may need to believe a conjecture of Riche and Williamson to define)" what do you mean? Didn't you and Catharina construct this over any field? $\endgroup$ – Chris Bowman Apr 14 '16 at 14:50
  • $\begingroup$ @ChrisBowman I think that only works in type A. I assume you want an arbitrary reductive algebraic group. $\endgroup$ – Ben Webster Apr 15 '16 at 1:11

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