Minimal-dimensional representations of finite $W$-superalgebras and attainability of the super Kac–Weisfeiler bounds (Zeng–Shu Conjecture 0.3)

For a basic Lie superalgebra $\mathfrak{g}$ over $\mathbb{C}$ and a nilpotent element $e\in\mathfrak{g}_{\bar{0}}$, Zeng and Shu conjectured (Conjecture 0.3 of their PRIMS 2017 paper) that the finite $W$-superalgebra $U(\mathfrak{g},e)$ always affords a representation of the smallest possible dimension: a one-dimensional representation when $d_1=\dim\mathfrak{g}_{\bar{1}}-\dim\mathfrak{g}^{e}_{\bar{1}}$ is even, and a two-dimensional representation when $d_1$ is odd, in which case one-dimensional representations provably do not exist. This is the super analogue of Premet's conjecture on one-dimensional representations of finite $W$-algebras, and it is exactly the ingredient needed to make the super Kac–Weisfeiler lower bounds $p^{d_0/2}2^{\lfloor d_1/2\rfloor}$ for irreducible modules of basic Lie superalgebras in sufficiently large characteristic $p$ attained for every $p$-character. The conjecture is confirmed in the literature only for type A (all nilpotent orbits), for minimal nilpotent orbits in all basic types, and for regular nilpotent elements of $\mathfrak{osp}(1|2n)$; the general case — arbitrary nilpotent orbits in $\mathfrak{osp}(M|2n)$, $\mathfrak{D}(2|1;\alpha)$, $\mathfrak{F}(4)$, $\mathfrak{G}(3)$, and the non-regular orbits of $\mathfrak{osp}(1|2n)$ — is open, and no counterexample is known. The problem asks for a proof or refutation of the conjecture in full generality; an affirmative resolution would yield, as a consequence, the unconditional attainability theorem (Theorem 0.4 of the source paper).

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 compatible with the grading; elements of $\mathfrak{g}_{\bar{0}}$ and $\mathfrak{g}_{\bar{1}}$ are called even and odd. The basic Lie superalgebras are the finite-dimensional Lie superalgebras in Kac's classification whose even part is reductive and which carry an even, non-degenerate, supersymmetric, invariant bilinear form: the general and special linear series $\mathfrak{gl}(M|N)$ and $\mathfrak{sl}(M|N)$, the orthosymplectic series $\mathfrak{osp}(M|2n)$, and the exceptional families $\mathfrak{D}(2|1;\alpha)$, $\mathfrak{F}(4)$, $\mathfrak{G}(3)$.

Fix a nilpotent element $e$ of the even part $\mathfrak{g}_{\bar{0}}$ (nilpotent meaning that its adjoint action on $\mathfrak{g}$ is a nilpotent linear map), and write $\mathfrak{g}^{e}$ for its centralizer. Two integers control the problem: $d_0=\dim\mathfrak{g}_{\bar{0}}-\dim\mathfrak{g}^{e}_{\bar{0}}$ and $d_1=\dim\mathfrak{g}_{\bar{1}}-\dim\mathfrak{g}^{e}_{\bar{1}}$, the super-dimension halves of the orbit of $e$. One can choose an integer grading of $\mathfrak{g}$ with $e$ in degree $2$ (a good grading); the parity of $d_1$ then coincides with the parity of $\dim\mathfrak{g}_{(-1)\bar{1}}$, the odd dimension of the degree-$(-1)$ piece. This 'judging parity' turns out to govern the structure of the $W$-superalgebra attached to $e$.

The finite $W$-superalgebra $U(\mathfrak{g},e)$ is a filtered associative superalgebra over $\mathbb{C}$ quantizing the coordinate ring of the Slodowy slice through $e$, the transverse slice to the orbit of $e$ inside $\mathfrak{g}$. Concretely, with $\chi=(e,\cdot)$ under the invariant form and $Q_{\chi}$ the generalized Gelfand–Graev module $U(\mathfrak{g})/I$, where $I$ is the left ideal generated by $\{a-\chi(a)\mid a\in\mathfrak{m}\}$ for a suitable isotropic subspace $\mathfrak{m}$ adapted to $\chi$, one sets $U(\mathfrak{g},e)=(\operatorname{End}_{\mathfrak{g}}Q_{\chi})^{op}$. The PBW theorem for these algebras (Zeng–Shu, J. Algebra 438 (2015)) states that the associated graded algebra is a polynomial algebra on $\mathfrak{g}^{e}$ when $\dim\mathfrak{g}_{(-1)\bar{1}}$ is even, and acquires one extra exterior generator $\Theta$ when it is odd — the first manifestation of the parity dichotomy.

The motivation comes from modular representation theory. Over an algebraically closed field of characteristic $p>2$, the representations of the restricted Lie superalgebra $\mathfrak{g}_{\Bbbk}$ are organized by its $p$-characters $\xi\in(\mathfrak{g}_{\Bbbk}^{*})_{\bar{0}}$ through the reduced enveloping algebra $U_{\xi}(\mathfrak{g}_{\Bbbk})$, the quotient of the enveloping superalgebra by the relations $a^{p}-a^{\langle p\rangle}=\xi(a)$ on even elements. For a reductive Lie algebra, the classical Kac–Weisfeiler conjecture, proved by Premet, asserts that every irreducible $U_{\xi}(\mathfrak{g})$-module has dimension divisible by $p^{(\dim\mathfrak{g}-\dim\mathfrak{g}^{\bar{x}})/2}$ and that this bound is sharp. Premet connected the sharpness to finite $W$-algebras (the non-super analogue of $U(\mathfrak{g},e)$): a one-dimensional representation of the characteristic-zero $W$-algebra yields, after reduction modulo $p$, an irreducible module of the minimal dimension for $p\gg0$. Premet's conjecture that every finite $W$-algebra over $\mathbb{C}$ admits a one-dimensional representation was settled through the work of Premet and Losev.

In the super setting, Wang and Zhao formulated the super Kac–Weisfeiler property: for a basic Lie superalgebra in characteristic $p>2$ with $p$-character $\xi$, every irreducible $U_{\xi}(\mathfrak{g}_{\Bbbk})$-module has dimension divisible by $p^{d_0/2}2^{\lfloor d_1/2\rfloor}$, a mixture of $p$-power and $2$-power reflecting the even and odd parts; they proved this divisibility for all basic superalgebras. Whether the bound is attained — whether modules of exactly this minimal dimension always exist — is the attainability side. Zeng and Shu reduced it, along Premet's lines, to a characteristic-zero question, with genuinely new parity phenomena: the super Schur lemma says the endomorphism ring of an irreducible module over a superalgebra is either $\mathbb{C}$ (module of type $M$) or a Clifford algebra on one generator (type $Q$), and a type-$Q$ module is necessarily at least two-dimensional. When $d_1$ is odd, $U(\mathfrak{g},e)$ has no one-dimensional representations at all, and the conjectural minimal module is a two-dimensional module of type $Q$, tied to one-dimensional representations of the refined $W$-superalgebra $W'_{\chi}$, a subalgebra of $U(\mathfrak{g},e)$ absorbing the exterior generator $\Theta$. Conjecture 0.3 of Zeng–Shu — the super analogue of Premet's conjecture — packages the needed statement: $U(\mathfrak{g},e)$ affords a one-dimensional representation when $d_1$ is even and a two-dimensional representation when $d_1$ is odd. Under it, their attainability theorem (Theorem 0.4) shows the super Kac–Weisfeiler bounds are attained for every basic superalgebra, every $p$-character, and $p\gg0$. The conjecture is currently known for all nilpotent orbits of $\mathfrak{gl}(M|N)$ and $\mathfrak{sl}(M|N)$ (where $d_1$ is automatically even), for regular nilpotent elements of $\mathfrak{osp}(1|2n)$ (the first odd case), and for minimal nilpotent elements of every basic Lie superalgebra.

Problem Statement

Let $\mathfrak{g}$ be a basic Lie superalgebra over $\mathbb{C}$, let $e\in\mathfrak{g}_{\bar{0}}$ be nilpotent, let $\mathfrak{g}^{e}$ be the centralizer of $e$, and set $d_i=\dim\mathfrak{g}_{\bar{i}}-\dim\mathfrak{g}^{e}_{\bar{i}}$ for $i=0,1$. Let $U(\mathfrak{g},e)$ be the finite $W$-superalgebra associated with the pair $(\mathfrak{g},e)$.

(i) When $d_1$ is even, must $U(\mathfrak{g},e)$ afford a one-dimensional representation?

(ii) When $d_1$ is odd — in which case one-dimensional representations never exist — must $U(\mathfrak{g},e)$ afford a two-dimensional representation?

Prove or refute these statements for all basic Lie superalgebras and all nilpotent elements. An affirmative resolution makes Theorem 0.4 of the source paper unconditional: for every basic Lie superalgebra $\mathfrak{g}_{\Bbbk}$ over an algebraically closed field of sufficiently large characteristic $p$ and every $p$-character $\xi\in(\mathfrak{g}_{\Bbbk}^{*})_{\bar{0}}$, the reduced enveloping algebra $U_{\xi}(\mathfrak{g}_{\Bbbk})$ admits an irreducible representation of dimension exactly

$$p^{d_0/2}\,2^{\lfloor d_1/2\rfloor}$$

where $d_0=\dim(\mathfrak{g}_{\Bbbk})_{\bar{0}}-\dim(\mathfrak{g}_{\Bbbk}^{\bar{x}})_{\bar{0}}$ and $d_1=\dim(\mathfrak{g}_{\Bbbk})_{\bar{1}}-\dim(\mathfrak{g}_{\Bbbk}^{\bar{x}})_{\bar{1}}$ for an element $\bar{x}\in(\mathfrak{g}_{\Bbbk})_{\bar{0}}$ corresponding to $\xi$ under the invariant bilinear form.

References

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