Maximal atypical block equivalence for $\mathfrak{osp}(2k+2|2k)$ versus $\mathfrak{osp}(2k+1|2k)$

A question left open by Gruson and Serganova's geometric determination of orthosymplectic characters (Proc. Lond. Math. Soc. 2010) remains unsolved: their closing conjecture that the maximal atypical blocks of the Lie superalgebras $\mathfrak{osp}(2k+2|2k)$ and $\mathfrak{osp}(2k+1|2k)$, whose weight-diagram move graphs and character matrices provably coincide, are equivalent as categories — a 2022 expert survey (Gorelik–Hoyt–Serganova–Sherman) still lists this as an important open problem, and only a bijection between simple modules compatible with the Duflo–Serganova functor is known.

Background

A Lie superalgebra is a $\mathbb{Z}_2$-graded vector space $\mathfrak{g}=\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_{\bar 1}$ with a bracket respecting the grading; the basic classical Lie superalgebras are those carrying an even nondegenerate invariant bilinear form with reductive even part, and the two main series are the general linear $\mathfrak{gl}(m|n)$ and the orthosymplectic $\mathfrak{osp}(m|2n)$, whose even part is $\mathfrak{so}(m)\oplus\mathfrak{sp}(2n)$. The character problem — give a closed formula for the character $\mathrm{Ch}(L_\lambda)$ of every finite-dimensional simple module — was solved for $\mathfrak{gl}(m|n)$ by flag-supervariety geometry in 1996 (and later by purely algebraic methods), and for the orthosymplectic case by the source paper (Gruson–Serganova, 2010).

The invariant controlling difficulty is atypicality. The defect of $\mathfrak{osp}(m|2n)$ is $\min(\lfloor m/2\rfloor,n)$, the maximal size of a set of mutually orthogonal odd (isotropic) roots; a simple module is atypical of degree $k$ if $\lambda+\rho$ is orthogonal to $k$ mutually orthogonal odd roots, and maximally atypical when $k$ equals the defect. Modules split into blocks — indecomposable subcategories, parametrized here by central characters — and Gruson–Serganova's Theorem 2 shows that any block of atypicality $k$ is equivalent, via translation functors (tensor with the natural module and project to a block), to the principal block, i.e. the maximal atypical block containing the trivial module, of one of $\mathfrak{gl}(k|k)$, $\mathfrak{osp}(2k+1|2k)$ (the odd-$m$ or 'B-series', labeled $t=1$), or $\mathfrak{osp}(2k|2k)$ / $\mathfrak{osp}(2k+2|2k)$ (the even-$m$ or 'D-series', labeled $t=0$ or $t=2$ according to the core type).

Their bookkeeping is the weight diagram: an infinite half-line of positions labeled by $\times$ (one per unit of atypicality), the core symbols $<$ and $>$ recording the typical part of the central character, and empty positions. For principal-block weights the core sits at the 'tail position', and switching the tail symbol $>$ to $<$ converts the diagrams of the $\mathfrak{osp}(2k+2|2k)$ principal block into those of the $\mathfrak{osp}(2k+1|2k)$.

The geometric method computes, for parabolic subgroups $P$ of the orthosymplectic supergroup $G$ and $P$-dominant weights $\lambda$, the induced-module cohomology $\Gamma^i(G/P,\mathcal{L}_\lambda)$, and packages graded multiplicities of simples $L_\mu$ in the Poincaré polynomial $K^{\lambda,\mu}_{G,P}(z)=\sum_{i\ge 0}\,[\Gamma^i(G/P,\mathcal{L}_\lambda):L_\mu]\,z^i .$ Their Euler-characteristic formula expresses $\sum_\mu K^{\lambda,\mu}_{G,P}(-1)\mathrm{Ch}(L_\mu)$ in closed form via classical Borel–Weil–Bott theory (the Borel–Weil–Bott theorem computes cohomology of line bundles on the ordinary flag variety of the even part), and their Theorem 1 makes these numbers recursive in the parabolic through a Leray spectral sequence. The resulting character formula writes $\mathrm{Ch}(L_\lambda)$ as an alternating path count over 'legal and exceptional moves' of weight diagrams applied to explicitly computable Euler characteristics $E_\mu$. Full polynomials $K^{\lambda,\mu}_{G,P^1}(z)$ are obtained only for the minimal parabolic $P^1$ in the maximal atypical block, and only Euler values $K^{\lambda,\mu}_{G,Q_\lambda}(-1)$ for the admissible parabolic $Q_\lambda$ attached to the 'tail subalgebra' of $\lambda$; the paper itself notes that for $G/B$ it is 'only able to calculate the Euler characteristic'.

In a §12 remark the authors observe that the move graph $D_{\mathfrak{g}}$, and hence the matrices $K(-1)$ and their inverses $D$ governing the character formula, are identical for $\mathfrak{osp}(2k+2|2k)$ and $\mathfrak{osp}(2k+1|2k)$ once diagrams are identified by the tail switch, and they conjecture that the maximal atypical blocks in the two cases are equivalent. The contrast is sharp: within the odd series the paper proves equivalences of maximal blocks (e.g. the principal blocks of $\mathfrak{osp}(2k+1|2k+2)$ and $\mathfrak{osp}(2k+1|2k)$ are equivalent), but no known functor crosses the $t=1$/$t=2$ boundary.

Later work sharpens the picture without closing it. The Duflo–Serganova cohomology functor $\mathrm{DS}_x$ sends a $\mathfrak{g}$-module to the homology of an odd element $x$ with $[x,x]=0$, regarded as a module for the centralizer of $x$; Gorelik–Heidersdorf (2022–2023) proved Serganova's conjecture that $\mathrm{DS}_x$ is semisimple on $\mathfrak{osp}(m|2n)$, rederived the Gruson–Serganova character formula through it, and established an explicit bijection $\tau$ between the simple modules of the two principal blocks that preserves all $\mathrm{DS}$-multiplicities — but no equivalence of categories. Ehrig–Stroppel (2013) constructed a Koszul diagram algebra $H^\infty_k/I$ from perverse sheaves on isotropic Grassmannians and conjectured that every block of atypicality $k$ of the orthosymplectic supergroup is equivalent to its module category, which would imply the Gruson–Serganova conjecture; this too remains open outside small rank.

Problem Statement

The problem is the following question left open by Gruson–Serganova (2010).

Block-equivalence conjecture. Prove or disprove: for every $k\ge 1$, the maximal atypical block containing the trivial module of $\mathfrak{osp}(2k+2|2k)$ and that of $\mathfrak{osp}(2k+1|2k)$ are equivalent as abelian $\mathbb{C}$-linear categories respecting the parity-shift functor $\Pi$, the simple modules being identified by switching the tail core symbol in the weight diagram — the bijection $\tau$ later made explicit by Gorelik–Heidersdorf. Strengthened forms of independent interest: an equivalence commuting with the Duflo–Serganova functor $\operatorname{DS}_x$, or an equivalence of graded (Koszul) categories as predicted by the Ehrig–Stroppel diagram algebra $H^\infty_k/I$.

References

1. Caroline Gruson, Vera Serganova, Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras, Proceedings of the London Mathematical Society (3) 101 (2010), No. 3, 852–892, DOI 10.1112/plms/pdq014, arXiv:0906.0918, https://doi.org/10.1112/plms/pdq014
2. Maria Gorelik, Thorsten Heidersdorf, Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor, Journal für die reine und angewandte Mathematik (Crelles Journal), published online 4 April 2023, DOI 10.1515/crelle-2022-0080, arXiv:2104.12634, https://doi.org/10.1515/crelle-2022-0080
3. Maria Gorelik, Thorsten Heidersdorf, Semisimplicity of the DS functor for the orthosymplectic Lie superalgebra, Advances in Mathematics 394 (2022), Article 108012, DOI 10.1016/j.aim.2021.108012, arXiv:2010.14975, https://doi.org/10.1016/j.aim.2021.108012
4. Maria Gorelik, Crystal Hoyt, Vera Serganova, Alexander Sherman, The Duflo–Serganova Functor, Vingt Ans Après, Journal of the Indian Institute of Science 102 (2022), No. 3, 961–1000, DOI 10.1007/s41745-022-00334-9, arXiv:2203.00529, https://doi.org/10.1007/s41745-022-00334-9
5. 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
6. Shun-Jen Cheng, Ngau Lam, Weiqiang Wang, Super duality and irreducible characters of ortho-symplectic Lie superalgebras, Inventiones mathematicae 183 (2011), No. 1, 189–224, DOI 10.1007/s00222-010-0277-4, arXiv:0911.0129, https://doi.org/10.1007/s00222-010-0277-4
7. Shun-Jen Cheng, Jae-Hoon Kwon, Kac–Wakimoto character formula for ortho-symplectic Lie superalgebras, Advances in Mathematics 304 (2017), 1296–1329, DOI 10.1016/j.aim.2016.09.026, arXiv:1406.6739, https://doi.org/10.1016/j.aim.2016.09.026
8. Caroline Gruson, Vera Serganova, Category of Finite Dimensional Modules over an Orthosymplectic Lie Superalgebra: Small Rank Examples, in: Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statistics (2013), 155–173, DOI 10.1007/978-1-4471-4863-0_7, https://doi.org/10.1007/978-1-4471-4863-0_7
9. Caroline Gruson, Vera Serganova, Bernstein-Gel'fand-Gel'fand reciprocity and indecomposable projective modules for classical algebraic supergroups, Moscow Mathematical Journal 13 (2013), No. 2, 281–313, DOI 10.17323/1609-4514-2013-13-2-281-313, arXiv:1111.6959, https://doi.org/10.17323/1609-4514-2013-13-2-281-313
10. Steven V. Sam, Borel–Weil factorization for super Grassmannians, Selecta Mathematica (New Series) 32 (2026), No. 2, Article 24, DOI 10.1007/s00029-026-01131-3, arXiv:2407.05167, https://doi.org/10.1007/s00029-026-01131-3