Super first Kac–Weisfeiler conjecture for finite-dimensional restricted Lie superalgebras

For a finite-dimensional restricted Lie superalgebra $\mathfrak g=\mathfrak g_{\bar 0}\oplus\mathfrak g_{\bar 1}$ over an algebraically closed field of characteristic $p>2$, every irreducible module is finite-dimensional, carries a $p$-character $\chi\in\mathfrak g_{\bar 0}^{*}$, and factors through a reduced enveloping algebra $U_\chi(\mathfrak g)$. Bin Shu (2024) proposed, as a super analogue of the 1971 first Kac–Weisfeiler conjecture, that the maximal dimension of an irreducible $\mathfrak g$-module equals $\mathscr M(\mathfrak g)=\max_{\chi}p^{b_0^\chi/2}2^{\lfloor b_1^\chi/2\rfloor}$, where $b_i^\chi=\dim\mathfrak g_{\bar i}-\dim\mathfrak z_{\bar i}^\chi$ and $\mathfrak z^\chi=\{X:\chi([X,\mathfrak g])=0\}$. The conjecture is proved only for basic classical, completely solvable, and strange classical (queer and periplectic) Lie superalgebras, and remains open for arbitrary restricted Lie superalgebras; its purely even case is the ordinary first Kac–Weisfeiler conjecture, itself still open in full generality. A companion question of the same paper asks whether the Wang–Zhao super KW divisibility property holds for every finite-dimensional solvable Lie superalgebra.

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 satisfying graded antisymmetry and the graded Jacobi identity; the even part $\mathfrak g_{\bar 0}$ is an ordinary Lie algebra and $\mathfrak g_{\bar 1}$ is its module. Over an algebraically closed field $\mathbf k$ of characteristic $p>2$, $\mathfrak g$ is called restricted if $\mathfrak g_{\bar 0}$ carries a $p$-th power map $[p]$ compatible with the bracket (a $p$-mapping, making $\mathfrak g_{\bar 0}$ a restricted Lie algebra in the sense of Jacobson). For such $\mathfrak g$, each irreducible module $V$ is finite-dimensional and admits a linear $p$-character $\chi\in\mathfrak g_{\bar 0}^{*}$ with $x^p-x^{[p]}$ acting as $\chi(x)^p$; $V$ is then a module over the reduced enveloping algebra $U_\chi(\mathfrak g)$, the quotient of the enveloping algebra by the relations $x^p-x^{[p]}-\chi(x)^p$ for $x\in\mathfrak g_{\bar 0}$.

The prototype is the first Kac–Weisfeiler conjecture for restricted Lie algebras (Weisfeiler–Kac, 1971). For a restricted Lie algebra $\mathfrak g$ and $\chi\in\mathfrak g^{*}$, let $\mathfrak g_\chi=\{x:\chi([x,\mathfrak g])=0\}$ be the centralizer of $\chi$ with respect to the bilinear form $(x,y)\mapsto\chi([x,y])$; the index of $\mathfrak g$ is $\operatorname{ind}\mathfrak g=\min_\chi\dim\mathfrak g_\chi$. The conjecture predicts that the maximal dimension of a simple $\mathfrak g$-module is $p^{(\dim\mathfrak g-\operatorname{ind}\mathfrak g)/2}$, equivalently that dimensions of simple modules at $\chi$ are governed by $\dim\mathfrak g-\dim\mathfrak g_\chi$. Premet proved it for Lie algebras of reductive algebraic groups (1995); Martin, Stewart, Tikaradze and Topley proved it for restricted Lie subalgebras of $\mathfrak{gl}_n$ in characteristics $p\gg n$ (2019), and Tikaradze gave a short proof for Lie algebras of group schemes over finitely generated subrings of $\mathbb C$ after base change to large positive characteristic. The conjecture remains open for arbitrary restricted Lie algebras in fixed small characteristics, and Topley (2017) constructed a non-restricted Lie algebra where the prediction fails, showing that restrictedness is essential.

Wang and Zhao (2009) initiated the modular representation theory of restricted Lie superalgebras. They formulated the Super KW Property: for a restricted Lie superalgebra $\mathfrak g$, $\chi\in\mathfrak g_{\bar 0}^{*}$, and $d_i=\dim\mathfrak g_{\bar i}-\dim\mathfrak g_{\chi,\bar i}$ with $\mathfrak g_\chi=\{y:\chi([y,\mathfrak g])=0\}$, the dimension of every irreducible $U_\chi(\mathfrak g)$-module is divisible by $p^{d_0/2}2^{\lfloor d_1/2\rfloor}$ — a mixture of $p$-power divisibility (from the even part) and $2$-power divisibility (from the odd part, where brackets are symmetric). They proved this for all basic classical Lie superalgebras (those with a nondegenerate even supersymmetric form and reductive even part: $\mathfrak{gl}(m|n)$, $\mathfrak{sl}(m|n)$, $\mathfrak{osp}(m|n)$, and the exceptional ones, with restrictions on $p$) and asked for which Lie superalgebras the property holds. Zeng and Shu (2016) constructed Kac–Weisfeiler modules, i.e. irreducible modules attaining this divisibility bound, for $\mathfrak{gl}(m|n)$ and $\mathfrak{sl}(m|n)$.

Shu (2024) proposed the corresponding maximal-dimension statement, Conjecture 2.3, as the super version of the first Kac–Weisfeiler conjecture, and proved it for all basic classical and all completely solvable restricted Lie superalgebras (completely solvable means the derived superalgebra $\mathfrak g^{(1)}$ is nilpotent). Ren, Shu, Yang and Zhang verified it for the strange classical Lie superalgebras — the queer series $\mathfrak q(n)$ and the periplectic series $\mathfrak p(n)$, the two classical families beyond the basic ones, with even parts $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ respectively and no nondegenerate even invariant bilinear form, which is why they lie outside the basic classical list. In the same paper, a theorem on the structure of irreducible modules of general solvable Lie superalgebras (Theorem 4.7) was corrected by a corrigendum (2026), an isolated fix that does not affect the conjecture. Two questions from this paper remain open and form the present problem: the conjecture for arbitrary finite-dimensional restricted Lie superalgebras, and whether the Super KW Property holds for every finite-dimensional solvable Lie superalgebra.

Problem Statement

The problem is to settle, by proof or explicit counterexample, the two open questions raised in Shu's paper on the super first Kac–Weisfeiler conjecture, extending the Kac–Weisfeiler dimension theory of irreducible modules beyond the classes where it is established (basic classical, completely solvable, and strange classical of queer/periplectic type): the maximal-dimension conjecture for every finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf k$ of characteristic $p>2$, and the Wang–Zhao divisibility question for every finite-dimensional solvable Lie superalgebra over $\mathbf k$.

(A) Super first Kac–Weisfeiler conjecture (Shu's Conjecture 2.3), general case. For $\chi\in\mathfrak g_{\bar 0}^{*}$ define the centralizer $\mathfrak z^\chi=\{X\in\mathfrak g:\chi([X,\mathfrak g])=0\}$ (equivalently the radical of the bilinear form $\mathsf B_\chi(X,Y)=\chi([X,Y])$), set $b_0^\chi=\dim\mathfrak g_{\bar 0}-\dim\mathfrak z_{\bar 0}^\chi$ and $b_1^\chi=\dim\mathfrak g_{\bar 1}-\dim\mathfrak z_{\bar 1}^\chi$, and let

$\mathscr M(\mathfrak g)=\max_{\chi\in\mathfrak g_{\bar 0}^{*}}\;p^{b_0^\chi/2}\,2^{\lfloor b_1^\chi/2\rfloor}.$

Is the maximal dimension of an irreducible $\mathfrak g$-module — equivalently, of irreducible $U_\chi(\mathfrak g)$-modules as $\chi$ ranges over all $p$-characters — equal to $\mathscr M(\mathfrak g)$ for every finite-dimensional restricted Lie superalgebra $\mathfrak g$? When $\mathfrak g_{\bar 1}=0$ this is exactly the ordinary first Kac–Weisfeiler conjecture, so the requested generality genuinely includes that open special case.

(B) Companion question (Shu's Remark 4.8(2)). Does the Wang–Zhao Super KW Property hold for every finite-dimensional solvable Lie superalgebra $\mathfrak g$ over $\mathbf k$? Since such a $\mathfrak g$ is solvable but not necessarily restricted, the property is read via its minimal restricted $p$-envelope $\mathfrak g_p=(\mathfrak g_{\bar 0})_p\oplus\mathfrak g_{\bar 1}$ (the smallest restricted Lie superalgebra containing $\mathfrak g$, in which $\mathfrak g$ is an ideal): every irreducible $\mathfrak g$-module extends to an irreducible $\mathfrak g_p$-module, which is associated with a unique $\Upsilon\in(\mathfrak g_p)_{\bar 0}^{*}$ whose restriction $\chi=\Upsilon|_{\mathfrak g_{\bar 0}}$ is the associated linear function on $\mathfrak g_{\bar 0}$. The question asks whether, for every finite-dimensional solvable $\mathfrak g$, every $\Upsilon\in(\mathfrak g_p)_{\bar 0}^{*}$, and every irreducible $U_\Upsilon(\mathfrak g_p)$-module $V$, $\dim V$ is divisible by $p^{b_0^\Upsilon/2}2^{\lfloor b_1^\Upsilon/2\rfloor}$, where $b_i^\Upsilon=\dim(\mathfrak g_p)_{\bar i}-\dim(\mathfrak z_p^\Upsilon)_{\bar i}$ with $\mathfrak z_p^\Upsilon=\{X\in\mathfrak g_p:\Upsilon([X,\mathfrak g_p])=0\}$, computed in $\mathfrak g_p$ exactly as in (A). (The Super KW Property is Wang–Zhao's divisibility statement as just formulated.) Shu proved the stronger statement that for completely solvable restricted $\mathfrak g$, all irreducible modules at a given $\chi$ have dimension exactly $p^{b_0^\chi/2}2^{\lfloor b_1^\chi/2\rfloor}$ (Theorem 5.5 of the source paper), so the divisibility property holds there; the open part is solvable-but-not-completely-solvable $\mathfrak g$, i.e. solvable $\mathfrak g$ whose derived superalgebra $\mathfrak g^{(1)}$ is not nilpotent.

References

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