Generalised Verma modules, closed character formulae, and cohomology for higher-rank orthosymplectic Lie superalgebras $\mathfrak{osp}_{k|2n}$

For the orthosymplectic Lie superalgebra $\mathfrak{osp}_{k|2}$, Su and Zhang (2012) determined the composition factors and multiplicities of the parabolic generalised Verma modules induced from the maximal parabolic subalgebra obtained by removing the unique odd simple root, derived from this closed finite-sum Weyl-type character and dimension formulae for the atypical finite-dimensional irreducible modules and Kac modules, and computed the first and second cohomology groups with coefficients in these modules. The paper explicitly limits itself to the rank-one odd case ($n=1$). The open problem is the extension of this elementary, module-theoretic program to $\mathfrak{osp}_{k|2n}$ with $n>1$: explicit composition-factor descriptions of the generalised Verma modules, closed character/dimension formulae for atypical irreducible and Kac modules, and explicit first and second cohomology groups $H^1$, $H^2$ for $n>1$ with finite-dimensional coefficients. Later literature gives a conceptual Kazhdan–Lusztig-theoretic solution of the irreducible character problem for all finite-dimensional $\mathfrak{osp}_{m|2n}$-modules via super duality, and closed formulae for special classes (tame modules, $\mathfrak{osp}(3|2m)$), but the elementary structural descriptions and the explicit first and second cohomology computations for general $n>1$ remain open.

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 graded bracket that is antisymmetric on the odd part; its finite-dimensional modules are graded, and the category of finite-dimensional modules over a classical Lie superalgebra is in general not semisimple. The orthosymplectic superalgebra $\mathfrak{osp}_{k|2n}$ is the Lie superalgebra of endomorphisms of the superspace $\mathbb{C}^{k|2n}$ preserving a nondegenerate even supersymmetric bilinear form; its even part is $\mathfrak{so}(k)\oplus\mathfrak{sp}(2n)$ and its odd part is the tensor product of the defining $\mathfrak{so}(k)$- and $\mathfrak{sp}(2n)$-modules. Unlike semisimple Lie algebras, a basic classical Lie superalgebra has non-conjugate Borel subalgebras. Fixing the distinguished Borel subalgebra (the one containing the largest possible number of even simple roots), the Dynkin diagram of $\mathfrak{osp}_{k|2n}$ contains exactly one odd simple root, and removing it yields a maximal parabolic subalgebra $\mathfrak{p}$. The generalised (parabolic) Verma modules studied by Su and Zhang are the modules $U(\mathfrak{g})\otimes_{U(\mathfrak{p})}E(\lambda)$ induced from finite-dimensional irreducible $\mathfrak{p}$-modules $E(\lambda)$; their maximal finite-dimensional quotients are the Kac modules, i.e. the modules induced from the irreducible module of the even part extended trivially over the positive odd nilradical, which play the role that Verma modules play in the purely even theory. A dominant integral weight $\lambda$ is typical if $(\lambda+\rho,\beta)\neq 0$ for every odd positive root $\beta$ (where $\rho$ is the super half-sum of positive roots); typical modules have Weyl-type characters by a theorem of Kac, while atypical modules do not, and finding their characters is one of the central problems of Lie superalgebra representation theory. In the $\mathfrak{osp}_{k|2}$ case ($n=1$) Su and Zhang determined exactly which irreducible modules occur in the generalised Verma modules and with what multiplicities, described the lattices of primitive weights (a primitive weight of a module is the highest weight of one of its nonzero submodules), which pin down the submodule structure, obtained from this finite-sum character and dimension formulae resembling Weyl's formulas for every finite-dimensional irreducible module and the Kac modules, and, using these structural results, computed the first and second Lie superalgebra cohomology groups $H^1(\mathfrak{g},M)$ and $H^2(\mathfrak{g},M)$ with $M$ ranging over finite-dimensional irreducible and Kac modules (Lie superalgebra cohomology is the derived functor of the invariants functor $M\mapsto M^{\mathfrak{g}}$). Three neighbouring frameworks existed alongside or appeared just after: the geometric Gruson–Serganova algorithm, which computes every finite-dimensional irreducible character of a basic classical superalgebra but without closed formulae; the super duality of Cheng, Lam and Wang, an equivalence between parabolic categories $\mathcal{O}$ (category $\mathcal{O}$ being the highest-weight category of finitely generated modules that are diagonalizable over the Cartan subalgebra and locally finite over the nilradical of the fixed Borel) of infinite-rank ortho-symplectic superalgebras and classical Lie algebras which, combined with truncation to finite rank, expresses all finite-dimensional irreducible $\mathfrak{osp}_{m|2n}$-characters (integral highest weights; non-integral ones are typical) through parabolic Kazhdan–Lusztig polynomials of BCD types — Kazhdan–Lusztig polynomials being the universal polynomials controlling composition multiplicities in category $\mathcal{O}$; and the Kac–Wakimoto character formula, a closed denominator-type formula proved by Cheng and Kwon for the special class of tame modules (atypical modules whose atypical odd roots are mutually orthogonal and each orthogonal to the shifted weight $\lambda+\rho$). Cheng and Kwon also record the field consensus that, beyond such special classes, closed character formulae for ortho-symplectic superalgebras are not expected in general, because the controlling Kazhdan–Lusztig-type polynomials are complicated to compute. What these frameworks do not provide is the elementary, module-theoretic picture for $n>1$: explicit composition-factor and submodule-structure descriptions of the generalised Verma modules in the style of the $n=1$ theory, closed finite-sum character formulae for atypical modules, and explicit cohomology groups. On the cohomology side, the known explicit computations before and after the source paper cover only $\mathfrak{sl}_{m|n}$ and $\mathfrak{osp}_{2|2n}$ (first and second cohomology, Su–Zhang 2007), $\mathfrak{osp}_{3|2}$ (Gruson 2003), and $\mathfrak{osp}_{k|2}$ itself; general structural tools such as the cohomological detection theorem of Lehrer, Nakano and Zhang and the support-variety interpretation of atypicality for $\mathfrak{osp}_{m|2n}$ by Kujawa (support varieties being cohomologically defined geometric invariants attached to modules, whose dimension detects atypicality) do not compute the groups explicitly.

Problem Statement

Let $\mathfrak{g}=\mathfrak{osp}_{k|2n}$ with $k\geq 3$ over $\mathbb{C}$, equipped with the distinguished Borel subalgebra, and let $\mathfrak{p}$ be the maximal parabolic subalgebra obtained by removing the unique odd simple root. The following questions extend the $n=1$ results of Su and Zhang (2012) to $n>1$:

1. (Structure) For each finite-dimensional irreducible $\mathfrak{p}$-module $E(\lambda)$, determine the composition factors of the generalised Verma module $U(\mathfrak{g})\otimes_{U(\mathfrak{p})}E(\lambda)$ with their multiplicities, in explicit elementary (combinatorial) form, together with the submodule structure via its primitive-weight lattice — the analogue for $n>1$ of the source paper's Theorem 3.16.

2. (Characters) Derive closed formulae for the formal characters and dimensions of the finite-dimensional irreducible $\mathfrak{g}$-modules and of the Kac modules with atypical highest weights, as finite sums of Weyl-type terms — the analogue of the source paper's Theorem 3.17 and Proposition 3.18 for $n>1$. Here "closed" means a genuinely new finite-sum Weyl-type formula valid for general $n>1$, as opposed to a restatement of the Kazhdan–Lusztig/super-duality expression or one of the already-solved cases (the tame modules, or $k=3$, i.e. $\mathfrak{osp}(3|2m)$); alternatively, show that no such finite-sum formula can hold in general and identify the obstruction.

3. (Cohomology) Compute explicitly the first and second Lie superalgebra cohomology groups $H^1(\mathfrak{g},L_\lambda)$, $H^2(\mathfrak{g},L_\lambda)$, $H^1(\mathfrak{g},K_\lambda)$ and $H^2(\mathfrak{g},K_\lambda)$ for general $n>1$, where $L_\lambda$ ranges over finite-dimensional irreducible modules and $K_\lambda$ over Kac modules — extending the source paper's cohomology theorem (Theorem 4.1 in arXiv:0911.0735v2).

The overarching objective is an elementary, module-theoretic theory of the higher-rank ortho-symplectic superalgebras supplying (1), (2) and (3) for general $n>1$.

References

1. Yucai Su and 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, https://doi.org/10.1016/j.jalgebra.2012.01.026 (arXiv:0911.0735, https://arxiv.org/abs/0911.0735)
2. Shun-Jen Cheng, Ngau Lam, and 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 (arXiv:0911.0129, https://arxiv.org/abs/0911.0129)
3. Caroline Gruson and 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 (arXiv:0906.0918, https://arxiv.org/abs/0906.0918)
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6. Shun-Jen Cheng and 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, https://doi.org/10.1016/j.aim.2016.09.026 (arXiv:1406.6739, https://arxiv.org/abs/1406.6739)
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8. Caroline Gruson, "Cohomologie des modules de dimension finie sur la super algèbre de Lie osp(3,2)", Journal of Algebra 259 (2003) 581-598, DOI 10.1016/S0021-8693(02)00573-2, https://doi.org/10.1016/S0021-8693(02)00573-2
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10. Jonathan Kujawa, "The generalized Kac-Wakimoto conjecture and support varieties for the Lie superalgebra osp(m|2n)", Proceedings of Symposia in Pure Mathematics 86 (2012) 201-215, DOI 10.1090/pspum/086/1419, https://doi.org/10.1090/pspum/086/1419 (arXiv:1112.3384, https://arxiv.org/abs/1112.3384)
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