Chuang–Miyachi's Koszul duality conjecture for affine category O blocks

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Let e,ℓ,m>0e,\ell,m>0, let ν∈Zℓ(m)\nu\in\mathbb{Z}^{\ell}(m), and let Oμ,aν\mathcal{O}^{\nu}_{\mu,a} denote the block defined by

Oμ,aν=Oμν(n),n=⟨aδ+∑p=1ℓωνp−μ:ρ^e⟩,\mathcal{O}_{\mu,a}^{\nu}=\mathcal{O}^{\nu}_{\mu}(n),\qquad n=\Bigl\langle a\delta+\sum_{p=1}^{\ell}\omega_{\nu_p}-\mu:\hat\rho_e\Bigr\rangle,

with a∈Na\in\mathbb{N}. Chuang–Miyachi's conjecture. The blocks Oμ,aν\mathcal{O}^{\nu}_{\mu,a} and Oν,aμ\mathcal{O}^{\mu}_{\nu,a} are Koszul and are Koszul dual to each other. This predicts a symmetry exchanging the affine parameters ee and ℓ\ell at the level of block categories; the source provides no resolution status.

References

Primary source

Peng Shan, Michela Varagnolo and Eric Vasserot, “Koszul duality of affine Kac-Moody algebras and cyclotomic rational DAHA”, arXiv:1107.0146 (2013).

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