Uniform closed character theory for finite-dimensional irreducible modules over the general ortho-symplectic Lie superalgebras (II-type, types B(m|n) and D(m|n))

For the general ortho-symplectic Lie superalgebras $\mathfrak{osp}(2m+1|2n)$ and $\mathfrak{osp}(2m|2n)$ — Kac's II-type basic classical Lie superalgebras of types B and D — no uniform closed character theory for finite-dimensional irreducible modules is known beyond the tame class. Every such character is computable by terminating algorithms (Gruson–Serganova's supergrassmannian-cohomology algorithm; Cheng–Lam–Wang's super duality, which reduces them to classical Kazhdan–Lusztig polynomials), and closed Weyl-type formulae exist for typical modules (Kac), for all tame modules (the Kac–Wakimoto conjecture, proved for osp by Cheng–Kwon), for atypicality one in the neighbouring series $\mathfrak{gl}(m|n)$ (Bernstein–Leites) and $\mathfrak{osp}(2|2n)$ (van der Jeugt), and for the rank-one sector families $\mathfrak{osp}(n|2)$ and $\mathfrak{osp}(3|2m)$ (Luo; Su–Zhang; Cao–Luo). The open core, stated by Luo in 2009 and still open per Cheng–Kwon (2017), is a uniform closed-form character theory — explicit closed formulae, or positive combinatorial descriptions of the governing Kazhdan–Lusztig-type polynomials — for the remaining non-tame atypical modules, with no restriction on the degree of atypicality.

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; its finite-dimensional modules carry a compatible grading, and the character $\mathrm{ch}\,L=\sum_\mu \dim L_\mu\, e^\mu$ records all weight multiplicities (each weight space being the direct sum of its even and odd parts), hence the dimension and the superdimension (the difference of the even and the odd dimensions). The basic classical Lie superalgebras are the finite-dimensional simple superalgebras with an even nondegenerate invariant bilinear form; besides the general linear series they comprise the ortho-symplectic series $\mathfrak{osp}(m|2n)$, whose even part is $\mathfrak{so}(m)\oplus\mathfrak{sp}(2n)$ and whose odd part is the natural $\mathfrak{so}(m)$–$\mathfrak{sp}(2n)$ bimodule, together with the exceptional series. Kac divided them into type I (admitting a consistent $\mathbb{Z}$-grading $\mathfrak{g}_{-1}\oplus\mathfrak{g}_{\bar 0}\oplus\mathfrak{g}_1$, e.g. $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(2|2n)$) and type II (no such grading: $\mathfrak{osp}(2m+1|2n)$ of type B$(m|n)$, $\mathfrak{osp}(2m|2n)$ of type D$(m|n)$, and the exceptional $D(2|1;\alpha)$, $F(4)$, $G(3)$); the type II structure is what makes their representation theory markedly harder.

Fix a Borel subalgebra and Cartan $\mathfrak{h}$. Weights are written in the standard coordinates $\epsilon_i$ (orthogonal) and $\delta_j$ (symplectic). A root $\alpha$ is isotropic if $(\alpha,\alpha)=0$ with respect to the invariant form — for osp these are exactly the odd roots $\pm\epsilon_i\pm\delta_j$ (the roots $\pm\delta_i\pm\delta_j$ of the symplectic factor are even and non-isotropic). A highest weight $\lambda$ is typical if $(\lambda+\rho,\alpha)\neq 0$ for every isotropic root $\alpha$, where $\rho=\rho_{\bar 0}-\rho_{\bar 1}$ is the difference of the even and odd half-sums of positive roots; otherwise it is atypical, and its degree of atypicality is the maximal number of mutually orthogonal isotropic positive roots orthogonal to $\lambda+\rho$. Kac (1977) proved a Weyl-type closed formula for typical finite-dimensional irreducibles of all basic classical superalgebras; the atypical modules have no such formula in general, and computing their characters has been the central open problem of the field since. A generalized Verma module is the parabolically induced module $M(\nu)$ whose character is computable by a Weyl-type alternating sum; expressing $\mathrm{ch}\,L(\lambda)$ as an explicit finite combination of such characters, with coefficients given by (generalized) Kazhdan–Lusztig polynomials — the polynomials that in Lie theory govern composition multiplicities of induced modules — is the standard form of a "complete character formula" in this subject. Kazhdan–Lusztig (KL) polynomials themselves are computable recursively but have closed forms only in low rank.

Known closed formulae before Luo (2009): Bernstein–Leites (1980, for $\mathfrak{gl}(m|n)$) and van der Jeugt (1991, for $\mathfrak{osp}(2|2n)$, the type-C border case) handled atypicality one; these were quickly seen not to extend to higher atypicality. Kac–Wakimoto (1994) conjectured a remarkable closed formula (a Weyl-type alternating Weyl-group sum in which the odd denominator is corrected only by a distinguished set $T_\lambda$ of $k$ mutually orthogonal isotropic odd simple roots, times a normalization $j_\lambda$) for the class of tame modules — those admitting a Borel subalgebra in whose simple roots the $k$ atypical roots of $L(\lambda)$ all appear as simple isotropic odd roots orthogonal to $\lambda+\rho$ — a class that can have atypicality exceeding one. For $\mathfrak{gl}(m|n)$ the full character problem was solved conceptually: Serganova's algorithm and Brundan's description via canonical bases (Lusztig-type distinguished bases of quantum-group modules with positive structure constants) give all characters, and Su–Zhang gave an explicit closed formula, which is the benchmark a "uniform closed theory" is measured against.

For type II the picture before Luo (2009) was: no complete character formula existed for any II-type basic classical superalgebra of types B and D; Luo then obtained one for the rank-one odd-sector family $\mathfrak{osp}(n|2)$ (types B$(m|1)$, D$(m|1)$, where the symplectic part $\mathfrak{sp}(2)$ has rank one), expressing every finite-dimensional irreducible character through generalized Verma module characters and cohomologically defined generalized KL polynomials, and posed the general problem. Su–Zhang independently completed $\mathfrak{osp}(k|2)$ via generalized Verma modules, and Cao–Luo later completed the rank-one even-sector family $\mathfrak{osp}(3|2m)$. Subsequently Gruson–Serganova (2010) gave a geometric algorithm computing every finite-dimensional simple character of every basic classical superalgebra via Euler characteristics (virtual alternating sums of induced characters) and the cohomology of generalized supergrassmannians (the $\mathbb{Z}_2$-graded analogues of Grassmannian varieties), and Cheng–Lam–Wang (2011) proved a super duality giving a complete conceptual solution of the irreducible character problem for osp in a parabolic category O containing all finite-dimensional irreducibles, in terms of classical KL polynomials. Thus all osp characters are effectively computable — but through algorithms and KL-polynomial recursions, not closed formulae. Cheng–Kwon (2017) then classified the tame highest weights for $\mathfrak{osp}(2m+1|2n)$ and $\mathfrak{osp}(2m|2n)$ (non-typical tame weights are exactly those of atypicality only of "type A") and proved the Kac–Wakimoto formula for all finite-dimensional tame osp modules, identifying the super characters up to sign with Sergeev–Veselov specialized super Jacobi polynomials. They emphasize that outside the tame subclass the finite-dimensional irreducible characters are controlled by intricate KL-type polynomials, that closed formulas are not expected in general, and that closed forms for non-tame characters remain a substantive bottleneck. Bao (2017) and Bao–Wang–Watanabe (2020) established the KL framework for the BGG category $\mathcal{O}$ (the highest-weight module category introduced by Bernstein–Gelfand–Gelfand) of $\mathfrak{osp}(2m|2n)$ and its parabolic variants — via quantum symmetric pairs (quantum-group analogues of symmetric pairs, equipped with i-canonical bases that play the role of Kazhdan–Lusztig bases) — a uniform reduction of all category-O characters to canonical-basis computations, but not to closed forms, and with no super-type-B analogue yet.

Problem Statement

Develop a uniform closed character theory for the finite-dimensional irreducible modules over the general II-type basic classical Lie superalgebras — the ortho-symplectic series $\mathfrak{osp}(2m+1|2n)$ (type $B(m|n)$) and $\mathfrak{osp}(2m|2n)$ (type $D(m|n)$), with no restriction on the degree of atypicality. Concretely: for every integral dominant highest weight $\lambda$ of these superalgebras, express the character $\mathrm{ch}\,L(\lambda)$ by an explicit closed formula — of Weyl–Kac, Bernstein–Leites, Kac–Wakimoto, or generalized-Verma type (a uniform finite explicit expansion in characters of generalized Verma modules or Euler characteristics whose coefficients are given in closed form or by a positive combinatorial description, not merely determined recursively or algorithmically as in Gruson–Serganova's supergrassmannian-cohomology algorithm or the super-duality reduction to classical Kazhdan–Lusztig polynomials) — extending the known formulae beyond the tame class; equivalently, provide explicit closed or positive combinatorial descriptions of the Kazhdan–Lusztig-type polynomials that govern these characters outside the tame class. Here a 'positive combinatorial description' is a closed or positive rule for the coefficients of the governing osp Kazhdan–Lusztig-type polynomials themselves — matching the explicitness of the closed formulae already known for $\mathfrak{gl}(m|n)$ and for the tame osp class — rather than a merely recursive or algorithmic determination, or a reduction of the osp characters to the classical Kazhdan–Lusztig polynomials of another algebra. The rank-one odd-sector family $\mathfrak{osp}(n|2)$, the rank-one even-sector family $\mathfrak{osp}(3|2m)$, and the tame class already possess such formulae; atypicality one alone does not guarantee a formula in the II-type series (atypicality-one formulae are known only for the neighbouring series $\mathfrak{gl}(m|n)$ and $\mathfrak{osp}(2|2n)$, and a non-typical II-type weight of atypicality one is tame only when its atypical root has the distinguished 'type A' form $\delta_i-\epsilon_j$, as follows from Cheng–Kwon's classification). The required theory must therefore cover all remaining finite-dimensional irreducible modules — the non-tame weights, of every degree of atypicality including one. This is the problem Luo posed in 2009 ('there is still no complete character formula for a II type basic classical Lie superalgebra (i.e. types B and D)') after settling $\mathfrak{osp}(n|2)$, read in the sense that remains open after the subsequent algorithmic solutions of Gruson–Serganova and Cheng–Lam–Wang and the tame-case resolution of Cheng–Kwon.

References

1. V. G. Kac, 'Lie superalgebras', Advances in Mathematics 26 (1977) 8–96. DOI: 10.1016/0001-8708(77)90017-2. https://doi.org/10.1016/0001-8708(77)90017-2
2. I. N. Bernstein, D. A. Leites, 'A formula for the characters of the irreducible finite-dimensional representations of Lie superalgebras of series gl and sl' (in Russian), C. R. Acad. Bulgare Sci. 33 (1980) 1049–1051. zbMATH: 0457.17002. https://zbmath.org/?q=an%3A0457.17002
3. J. van der Jeugt, 'Character formulae for the Lie superalgebra C(n)', Communications in Algebra 19 (1991) 199–222. DOI: 10.1080/00927879108824137. https://doi.org/10.1080/00927879108824137
4. V. G. Kac, M. Wakimoto, 'Integrable highest weight modules over affine superalgebras and number theory', in Lie Theory and Geometry, Progress in Mathematics 123, Birkhäuser, Boston (1994) 415–456. DOI: 10.1007/978-1-4612-0261-5_15. https://doi.org/10.1007/978-1-4612-0261-5_15
5. Yucai Su, R. B. Zhang, 'Character and dimension formulae for general linear superalgebra', Advances in Mathematics 211 (2007) 1–33. DOI: 10.1016/j.aim.2006.07.010. https://doi.org/10.1016/j.aim.2006.07.010
6. A. N. Sergeev, A. P. Veselov, 'Euler characters and super Jacobi polynomials', Advances in Mathematics 226 (2011) 4286–4315. DOI: 10.1016/j.aim.2010.11.015. arXiv:0905.2603. https://arxiv.org/abs/0905.2603
7. Caroline Gruson, Vera Serganova, 'Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras', Proceedings of the London Mathematical Society 101 (2010) 852–892. DOI: 10.1112/plms/pdq014. arXiv:0906.0918. https://arxiv.org/abs/0906.0918
8. 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
9. Li Luo, 'Character formulae for ortho-symplectic Lie superalgebras osp(n|2)', Journal of Algebra 353 (2012) 31–61. DOI: 10.1016/j.jalgebra.2011.12.014. arXiv:0909.3134. https://arxiv.org/abs/0909.3134
10. Bintao Cao, Li Luo, 'Generalized Verma Modules and Character Formulae for osp(3|2m)', arXiv:1001.3986 (2010). https://arxiv.org/abs/1001.3986
11. Yucai Su, R. B. Zhang, 'Generalised Verma modules for the orthosymplectic Lie superalgebra osp(k|2)', Journal of Algebra 357 (2012) 94–115. DOI: 10.1016/j.jalgebra.2012.01.026. arXiv:0911.0735. https://arxiv.org/abs/0911.0735
12. Michael Chmutov, Crystal Hoyt, Shifra Reif, 'The Kac–Wakimoto character formula for the general linear Lie superalgebra', Algebra & Number Theory 9 (2015) 1419–1452. DOI: 10.2140/ant.2015.9.1419. arXiv:1310.3798. https://arxiv.org/abs/1310.3798
13. Maria Gorelik, Victor Kac, 'Characters of (relatively) integrable modules over affine Lie superalgebras', Japanese Journal of Mathematics 10 (2015) 135–235. DOI: 10.1007/s11537-015-1464-2. arXiv:1406.6860. https://arxiv.org/abs/1406.6860
14. 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://arxiv.org/abs/1406.6739
15. Huanchen Bao, 'Kazhdan–Lusztig theory of super type D and quantum symmetric pairs', Representation Theory 21 (2017) 247–276. DOI: 10.1090/ert/505. arXiv:1603.05105. https://arxiv.org/abs/1603.05105
16. Chih-Whi Chen, Shun-Jen Cheng, Kevin Coulembier, 'Tilting modules for classical Lie superalgebras', Journal of the London Mathematical Society 103 (2021) 870–900. DOI: 10.1112/jlms.12395. arXiv:1907.06579. https://arxiv.org/abs/1907.06579
17. 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
18. Arun S. Kannan, Honglin Zhu, 'Characters for projective modules in the BGG category O for the orthosymplectic Lie superalgebra osp(3|4)', Journal of Algebra 569 (2021) 723–757. DOI: 10.1016/j.jalgebra.2020.10.030. arXiv:2006.06788. https://arxiv.org/abs/2006.06788
19. Maria Gorelik, Thorsten Heidersdorf, 'Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor', Journal für die reine und angewandte Mathematik 798 (2023) 1–54. DOI: 10.1515/crelle-2022-0080. arXiv:2104.12634. https://arxiv.org/abs/2104.12634