The super duality theorem of Cheng, Lam and Wang (Inventiones mathematicae, 2011) establishes an explicit equivalence between the parabolic Bernstein–Gelfand–Gelfand category O‾\overline{\mathcal{O}} of modules over ortho-symplectic Lie superalgebras and the corresponding category O\mathcal{O} of modules over classical Lie algebras at infinite rank, thereby expressing irreducible ortho-symplectic characters through classical Kazhdan–Lusztig polynomials. The authors' proof is purely algebraic, and they state that "the formulation of super duality suggests potential direct connections on the (super) geometric level behind the categories O\mathcal{O} and O‾\overline{\mathcal{O}}, which will be very important to develop." The algebraic extension to Kac–Moody superalgebras announced by the authors — together with the subsequent quantum and Whittaker developments of the program — has since been carried out in the literature, while the geometric agenda the authors call for remains open. The problem is to construct such a geometric (super-geometric) correspondence that underlies the equivalence O≅O‾\mathcal{O}\cong\overline{\mathcal{O}} and yields a geometric proof of super duality, in analogy with the role of Beilinson–Bernstein localization in the classical Kazhdan–Lusztig conjectures.

Background

A Lie superalgebra is a Z2\mathbb{Z}_2-graded vector space g=g0ˉ⊕g1ˉ\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_{\bar 1} with a graded bracket; the basic example here is the ortho-symplectic superalgebra osp(M∣2n)\mathfrak{osp}(M|2n), whose even part is so(M)⊕sp(2n)\mathfrak{so}(M)\oplus\mathfrak{sp}(2n) and whose odd part is the tensor representation of that even part. The Bernstein–Gelfand–Gelfand (BGG) category O\mathcal{O} consists of finitely generated, h\mathfrak{h}-diagonalizable modules locally finite over the positive Borel; its parabolic variant replaces the Borel by a parabolic subalgebra and its standard objects are parabolic Verma modules (induced from a simple module over the Levi factor). For a semisimple Lie algebra, O\mathcal{O} is a highest weight category (in the sense of Cline–Parshall–Scott: it has a duality and standard/costandard objects controlled by a partially ordered weight set), and the Kazhdan–Lusztig (KL) conjectures, proved by Beilinson–Bernstein and Brylinski–Kashiwara, express the multiplicity of a simple module in a Verma module through KL polynomials — combinatorially defined polynomials attached to pairs of Weyl group elements. The proofs go through localization: D-modules (modules over the sheaf of differential operators) on the flag variety G/BG/B, where parabolic category O\mathcal{O} is realized geometrically, and KL polynomials are reinterpreted via intersection cohomology of Schubert varieties (perverse sheaves). For Lie superalgebras this geometric machine has no direct analogue: Borel subalgebras are not conjugate (one can flip an odd simple root of zero self-pairing — an 'isotropic' root — between positive and negative by an odd reflection, thereby changing the chosen Borel), blocks of O\mathcal{O} (its indecomposable direct summands) may be non-finitary, containing infinitely many simple modules, and there is no single flag variety serving all choices. Super duality bypasses geometry algebraically: Cheng–Lam–Wang introduce infinite-rank Lie (super)algebras g⊂g~⊃g‾\mathfrak{g}\subset\widetilde{\mathfrak{g}}\supset\overline{\mathfrak{g}} with parabolic categories O,O~,O‾\mathcal{O},\widetilde{\mathcal{O}},\overline{\mathcal{O}}, functors T:O~→OT:\widetilde{\mathcal{O}}\to\mathcal{O} and T‾:O~→O‾\overline{T}:\widetilde{\mathcal{O}}\to\overline{\mathcal{O}} (both defined on the common auxiliary category), and prove that both are equivalences, so O≅O‾\mathcal{O}\cong\overline{\mathcal{O}} as highest weight categories; truncation functors trnk\mathrm{tr}^k_n then descend to every finite rank nn, and the functors match parabolic Vermas, simples, tilting modules (the self-dual modules admitting both a parabolic-Verma and a dual-parabolic-Verma filtration), and u\mathfrak{u}-homology groups (Lie algebra homology for the nilradical u\mathfrak{u} of the parabolic, the invariant-theoretic avatar of KL polynomials by Vogan's interpretation). This yields all irreducible osp\mathfrak{osp} characters, including all finite-dimensional ones, in terms of classical KL polynomials of types B, C, D. In their closing remarks the authors note that the classical KL conjectures rest on deep geometry (D-modules on flag manifolds) and that the formulation of super duality 'suggests potential direct connections on the (super) geometric level behind the categories O\mathcal{O} and O‾\overline{\mathcal{O}}, which will be very important to develop'; they also announce an adaptation to Kac–Moody superalgebras 'including affine superalgebras' with details to 'appear elsewhere'. Known partial neighbors on the geometric side: Ehrig–Stroppel identified endomorphism algebras of blocks of the finite-dimensional osp\mathfrak{osp} category with diagrammatic algebras connected to perverse sheaves on isotropic Grassmannians, which is geometry of the finite-dimensional category rather than of the super duality pair; and cohomological support-variety programs for category O\mathcal{O} of Lie superalgebras provide a different, cohomological geometry. The promised Kac–Moody/affine algebraic extension was subsequently established by Cheng–Kwon–Wang for a large class of contragredient superalgebras (including affine superalgebras, with the caveat that the KL-polynomial character formula does not reach the affine cases because the dual Lie algebra there is not of Kac–Moody type).

Problem Statement

Construct a direct geometric (super-geometric) underpinning of super duality, as called for in Cheng–Lam–Wang (Invent. Math. 183 (2011), Section 1.6): a mathematically rigorous correspondence that realizes, at the level of geometry, the equivalence between the parabolic BGG category O‾\overline{\mathcal{O}} of ortho-symplectic Lie superalgebra modules and the parabolic BGG category O\mathcal{O} of the corresponding classical Lie algebra modules established by super duality. Concretely, for the super duality setup of the source paper (or its established Kac–Moody/quantum generalizations), the geometric correspondence is to (1) identify geometric categories together with geometric functors realizing the super duality functors TT and T‾\overline{T} as equivalences of the infinite-rank categories and compatible with the truncation functors trnk\mathrm{tr}^k_n; and (2) prove the compatibility of this realization with the defining data of super duality, namely the correspondence of parabolic Verma modules, simple modules, and tilting modules, and the identification of the corresponding character or u\mathfrak{u}-homology data (Kazhdan–Lusztig polynomials in the finite-rank ortho-symplectic setup). Such a construction would constitute a geometric proof of super duality, playing the role that Beilinson–Bernstein localization and intersection cohomology of Schubert varieties play for the classical Kazhdan–Lusztig conjectures. A purely algebraic re-derivation of the equivalence, or a geometric model of an adjacent category alone (for example of the finite-dimensional category F\mathcal{F} of osp\mathfrak{osp} or of endomorphism algebras of its blocks, as in Ehrig–Stroppel), does not by itself answer the question; it must be the categories O\mathcal{O} and O‾\overline{\mathcal{O}} (or their established generalizations) that are connected through geometry.

References

  1. Shun-Jen Cheng, Ngau Lam, Weiqiang Wang, 'Super duality and irreducible characters of ortho-symplectic Lie superalgebras', Inventiones mathematicae 183 (2011), 189–224. DOI: 10.1007/s00222-010-0277-4, arXiv:0911.0129. https://arxiv.org/abs/0911.0129
  2. Shun-Jen Cheng, Jae-Hoon Kwon, Weiqiang Wang, 'Irreducible characters of Kac-Moody Lie superalgebras', Proceedings of the London Mathematical Society 110 (2015), 108–132. DOI: 10.1112/plms/pdu045, arXiv:1308.3030. https://arxiv.org/abs/1308.3030
  3. Shun-Jen Cheng, Ngau Lam, Weiqiang Wang, 'Super duality for general linear Lie superalgebras and applications', Proceedings of Symposia in Pure Mathematics 86 (2012), 113–136. DOI: 10.1090/pspum/086/1413, arXiv:1109.0667. https://arxiv.org/abs/1109.0667
  4. Michael Ehrig, Catharina Stroppel, 'Diagrams for perverse sheaves on isotropic Grassmannians and the supergroup SOSP(m|2n)', arXiv:1306.4043 (2013). https://arxiv.org/abs/1306.4043
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  6. Huanchen Bao, Weiqiang Wang, Hideya Watanabe, 'Canonical bases for tensor products and super Kazhdan–Lusztig theory', Journal of Pure and Applied Algebra 224 (2020), 106347. DOI: 10.1016/j.jpaa.2020.106347, arXiv:1808.09388. https://arxiv.org/abs/1808.09388
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  8. Shun-Jen Cheng, Weiqiang Wang, 'Super duality for Whittaker modules and finite W-algebras', Annals of Representation Theory 2 (2025), no. 4, 505–535. DOI: 10.5802/art.28, arXiv:2410.08617. https://arxiv.org/abs/2410.08617
  9. Chun-Ju Lai, Daniel K. Nakano, Arik Wilbert, 'Category O for Lie superalgebras', arXiv:2505.17563 (2025). https://arxiv.org/abs/2505.17563