Closed formula for Borel highest-weight conversion of finite-dimensional osp(2m|2n)-modules in the signed epsilon-delta sequence cases left open by Cheng–Kwon

For the ortho-symplectic Lie superalgebra osp(2m|2n), Cheng and Kwon reduced the computation of the highest weight of an arbitrary finite-dimensional irreducible module relative to an arbitrary Borel subalgebra to an explicit Frobenius-coordinate formula in all but one pair of cases: when the signed epsilon-delta sequence of the Borel ends in a delta and the sequence contains a sign (the s(b)=-1 case for the module L(lambda-natural), symmetrically s(b)=1 for its sigma-twist), where they state that no general formula is known and only a reduction via the type-D diagram automorphism is available. The problem asks for an explicit, uniform closed formula for these missing cases, or equivalently a proven uniform effective procedure yielding one, completing the highest-weight conversion table needed to apply the Kac–Wakimoto character formula relative to arbitrary Borels. Characters of non-tame modules are governed by Kazhdan–Lusztig type polynomials, for which no general closed forms are expected.

Background

A Lie superalgebra is a $\mathbb{Z}_2$-graded algebra whose even part is a Lie algebra and whose odd part carries an alternating bracket. The ortho-symplectic Lie superalgebra $\mathfrak{osp}(2m|2n)$ is the basic classical example of type D: its even part is $\mathfrak{so}(2m)\oplus\mathfrak{sp}(2n)$, and it preserves an even supersymmetric form on $\mathbb{C}^{2m|2n}$. Unlike Lie algebras, a basic Lie superalgebra has many non-conjugate Borel subalgebras (maximal solvable graded subalgebras): starting from the standard (distinguished) Borel $\mathfrak{b}^{st}$, one reaches any other Borel by a chain of odd reflections, each replacing an isotropic odd simple root $\beta$ (a root with $\langle\beta,\beta\rangle=0$) by $-\beta$, and the highest weight of a given irreducible module changes under such reflections by a rule that is non-uniform when the weight is orthogonal to $\beta$. For $\mathfrak{osp}(2m|2n)$, the Borels sharing the standard even part are parametrized by signed epsilon-delta ($\epsilon\delta$) sequences: words with $m$ letters $\epsilon$ and $n$ letters $\delta$ in which the right-most $\epsilon$ carries a sign $\pm$ precisely when the sequence ends in $\delta$ (cf. [1, Sec. 2.6]), and $s(\mathfrak{b})=-1$ precisely when that sign is a minus. By a theorem of Kac, the finite-dimensional irreducible integer-weight modules are $L(\lambda^{\natural})$ and $L(\lambda^{\natural}_{-})$, where $\lambda$ ranges over the $(n|m)$-hook partitions $H(n|m)=\{\lambda:\lambda_{n+1}\leq m\}$, and $\lambda^{\natural}$ is the standard weight shift $\lambda^{\natural}=\sum_{i=1}^{n}\lambda_i\delta_i+\sum_{j=1}^{m}\kappa_j\epsilon_j$ with $\kappa$ the transpose partition of $(\lambda_{n+1},\lambda_{n+2},\ldots)$, while $\lambda^{\natural}_{-}$ differs from $\lambda^{\natural}$ by the sign of the last $\epsilon$-coordinate. For a Borel $\mathfrak{b}$, $\lambda^{\mathfrak{b}}$ (respectively $\lambda^{\mathfrak{b}}_{-}$) denotes the $\mathfrak{b}$-highest weight of $L(\lambda^{\natural})$ (respectively $L(\lambda^{\natural}_{-})$), the data needed to write characters relative to $\mathfrak{b}$. The formula of Kac and Wakimoto (Transform. Groups 19 (2014), Conjecture 3.6) predicts that characters of the tame modules — those admitting a set of mutually orthogonal isotropic odd simple roots of cardinality equal to the degree of atypicality (the maximal number of mutually orthogonal isotropic odd roots orthogonal to the shifted highest weight) — are given by a closed Weyl-type determinant formula; typical (atypicality zero) modules are the simplest subclass. Chmutov–Hoyt–Reif proved this for $\mathfrak{gl}(m|n)$, and Cheng–Kwon (Adv. Math. 304 (2017)) classified the tame finite-dimensional $\mathfrak{osp}$-modules and proved the formula for them. Their classification of $\mathfrak{b}$-highest weights is computed by an explicit formula in the Frobenius coordinates of $\lambda$ (the arm and leg lengths along the diagonal of the Young diagram of $\lambda$) — their Eq. (2.10), built on super duality, an exact functor relating $\mathfrak{osp}$-modules to modules over infinite-dimensional limit algebras where semisimple character computations are available, developed by Cheng–Lam–Wang (Invent. Math. 183 (2011)) — in the cases where the $\epsilon\delta$-sequence ends in $\epsilon$, and ends in $\delta$ with $s(\mathfrak{b})=1$ for $\lambda^{\mathfrak{b}}$ (and, via the outer diagram automorphism $\sigma:\epsilon_m\mapsto-\epsilon_m$ with $L(\lambda^{\natural}_{-})\cong L(\lambda^{\natural})^{\sigma}$, ends in $\delta$ with $s(\mathfrak{b})=-1$ for $\lambda^{\mathfrak{b}}_{-}$). Their paper states verbatim that in the remaining cases no general formula for $\lambda^{\mathfrak{b}}$ (respectively $\lambda^{\mathfrak{b}}_{-}$) is known. Outside the tame class, no closed forms are known or expected: characters of arbitrary finite-dimensional irreducible $\mathfrak{osp}$-modules are computable by the Gruson–Serganova algorithm (Proc. Lond. Math. Soc. 101 (2010)) and are governed by super Kazhdan–Lusztig polynomials — analogues of the classical polynomials encoding composition multiplicities — as structured by super duality and quantum-symmetric-pair approaches (quantum symmetric pairs are coideal subalgebras of quantized enveloping algebras whose canonical bases play the role of Kazhdan–Lusztig bases) (Cheng–Kwon–Wang 2015; Bao–Wang 2018; Bao 2017; Bao–Wang–Watanabe 2020).

Problem Statement

For $\mathfrak{g}=\mathfrak{osp}(2m|2n)$ with $m\geq 2$ and $n\geq 1$, let $\lambda\in H(n|m)$ be a hook partition indexing the finite-dimensional irreducible integer-weight modules $L(\lambda^{\natural})$ and $L(\lambda^{\natural}_{-})$, and let $\mathfrak{b}$ be a Borel subalgebra with standard even part, corresponding to a signed $\epsilon\delta$-sequence with $m$ letters $\epsilon$ and $n$ letters $\delta$, where $s(\mathfrak{b})=-1$ iff the sequence carries at least one minus sign. Find an explicit general closed formula, uniform in $\lambda$ and the signed $\epsilon\delta$-sequence data, for the $\mathfrak{b}$-highest weight $\lambda^{\mathfrak{b}}$ — that is, the highest weight of the module $L(\lambda^{\natural})$ relative to $\mathfrak{b}$ — in the case the $\epsilon\delta$-sequence ends in $\delta$ and $s(\mathfrak{b})=-1$, and symmetrically for $\lambda^{\mathfrak{b}}_{-}$ (the $\mathfrak{b}$-highest weight of $L(\lambda^{\natural}_{-})$) when the sequence ends in $\delta$ and $s(\mathfrak{b})=1$. The formula should be of the same character as the Frobenius-coordinate formula valid in the remaining cases: it must specialize correctly to the known cases, apply for all $m\geq 2$, $n\geq 1$, $\lambda\in H(n|m)$ and all signed sequences of the two missing types, and be proven correct. Equivalently, prove a uniform effective procedure that outputs the weight in closed Frobenius-type coordinates, replacing the current case-by-case reduction through the diagram automorphism $\sigma$.

References

1. 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 (source of the open question)
2. Victor G. Kac, Minoru Wakimoto, Representations of affine superalgebras and mock theta functions, Transformation Groups 19 (2014) 383–455, DOI 10.1007/s00031-014-9263-z, arXiv:1308.1261, https://arxiv.org/abs/1308.1261
3. Michael Chmutov, Crystal Hoyt, Shifra Reif, 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
4. Maria Gorelik, Victor G. 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
5. 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, https://doi.org/10.1007/s00222-010-0277-4
6. 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, https://doi.org/10.1112/plms/pdq014
7. 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
8. Huanchen Bao, Weiqiang Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque 402 (2018), arXiv:1310.0103, https://arxiv.org/abs/1310.0103
9. Huanchen Bao, Kazhdan–Lusztig theory of super type D and quantum symmetric pairs, Representation Theory 21 (2017) 247–276, arXiv:1603.05105, https://arxiv.org/abs/1603.05105
10. 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
11. Maria Gorelik, Thorsten Heidersdorf, Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor, Journal für die reine und angewandte Mathematik (Crelle's Journal), published online 2023, DOI 10.1515/crelle-2022-0080, arXiv:2104.12634, https://arxiv.org/abs/2104.12634
12. Shunsuke Hirota, Categorification of generic Su–Zhang character formula, arXiv:2512.16095, https://arxiv.org/abs/2512.16095
13. Shunsuke Hirota, Some closed formulas for super Kazhdan–Lusztig polynomials, arXiv:2603.01390, https://arxiv.org/abs/2603.01390
14. Shunsuke Hirota, $\mathfrak{b}_1$-Verma $\mathfrak{b}_2$-dual Verma supermodules, arXiv:2510.25276, https://arxiv.org/abs/2510.25276