The super Kac-Weisfeiler divisibility property for the strange classical Lie superalgebras $q(n)$ and $\tilde p(n)$

Over an algebraically closed field of characteristic $p>2$ , the Super Kac-Weisfeiler Property predicts that for every p-character $\chi$ of a restricted Lie superalgebra $\mathfrak g$ , the dimension of every irreducible module over the reduced enveloping superalgebra $U_\chi(\mathfrak g)$ is divisible by $p^{d_0/2}2^{\lfloor d_1/2\rfloor}$ , where $d_i=\dim\mathfrak g_i-\dim\mathfrak g_{\chi,i}$ is the defect of the centralizer of $\chi$ . The property is known for the basic classical Lie superalgebras and for the queer series $q(n)$ with nilpotent p-characters, but, as Ren-Shu-Yang-Zhang (2023) note while proving the related maximal-dimension conjecture for the strange series, it remains unverified for $q(n)$ with arbitrary p-character and entirely open for the periplectic series $\tilde p(n)$ .

Background

A restricted Lie (super)algebra over an algebraically closed field $\mathbf k$ of characteristic $p>2$ carries a $p$ th-power map $x\mapsto x^{[p]}$ on its even part. Consequently, for every linear functional $\chi$ on the even part (a p-character), one can form the reduced enveloping (super)algebra $U_\chi(\mathfrak g)$ , the quotient of the universal enveloping algebra by the relations $x^p-x^{[p]}=\chi(x)$ for even $x$ ; every irreducible module of a finite-dimensional restricted Lie superalgebra factors through some $U_\chi(\mathfrak g)$ . For ordinary restricted Lie algebras, Kac and Weisfeiler conjectured in 1971 that when $\mathfrak g$ is the Lie algebra of a reductive group, the dimension of every irreducible $U_\chi(\mathfrak g)$ -module is divisible by $p^{(\dim\mathfrak g-\dim\mathfrak g_\chi)/2}$ , where $\mathfrak g_\chi$ is the centralizer of $\chi$ under the coadjoint pairing $x\mapsto\chi([x,\cdot])$ . Premet proved this in 1995, and Martin-Stewart-Topley (2019) extended the first Kac-Weisfeiler conjecture to all algebraic Lie algebras in sufficiently large characteristics, small characteristics remaining open. Wang and Zhao transferred the question to Lie superalgebras: they formulated the Super KW Property, which asserts that for a restricted Lie superalgebra $\mathfrak g=\mathfrak g_{\bar0}\oplus\mathfrak g_{\bar1}$ and a p-character $\chi\in\mathfrak g_{\bar0}^*$ (extended by zero on the odd part), the dimension of every $U_\chi(\mathfrak g)$ -module is divisible by $p^{d_0/2}2^{\lfloor d_1/2\rfloor}$ with $d_i=\dim\mathfrak g_i-\dim\mathfrak g_{\chi,i}$ . Here $d_0$ is always even, and the genuinely new 2-power factor reflects a Clifford-type degeneracy of the symmetric bilinear form $(X,Y)\mapsto\chi([X,Y])$ on the odd part, whose maximal isotropic subspaces need not have a unique parity. Wang and Zhao proved the property for the basic classical Lie superalgebras (those with a nondegenerate invariant even form and reductive even part, such as $\mathfrak{sl}(m|n)$ and $\mathfrak{osp}(m|2n)$ ), and Zhao later gave a new proof with an optimal assumption on $p$ . Zeng and Shu constructed Kac-Weisfeiler modules for the general and special linear superalgebras. The strange classical series of Kac's classification resists these techniques: the queer Lie superalgebra $q(n)$ has even part $\mathfrak{gl}(n)$ and odd part a second copy of $\mathfrak{gl}(n)$ mixed by the bracket, and the periplectic Lie superalgebra $\tilde p(n)$ consists of block matrices with even part $\mathfrak{gl}(n)$ , the upper-right block symmetric and the lower-left block skew-symmetric. Wang and Zhao (2011) proved the Super KW Property for $q(n)$ when $\chi$ is nilpotent, via a freeness theorem over a maximal isotropic subalgebra, and showed in their Remark 7.8 that the general case would follow from their Morita super-equivalence conjecture (Conjecture 7.3), which posits adjoint exact functors between $U_\chi(q(n))$ -mod and $U_\chi(q(n)_{\chi_s})$ -mod, where $\chi=\chi_s+\chi_n$ is the Jordan decomposition of the p-character into semisimple and nilpotent parts, $q(n)_{\chi_s}$ is the centralizer of $\chi_s$ in $q(n)$ , and $\chi$ in $U_\chi(q(n)_{\chi_s})$ is understood as its restriction to that centralizer; they verified this equivalence only for $q(2)$ (Theorem 7.5). For the periplectic series, essentially nothing was known before 2023. Shu (published 2024) then formulated the first super Kac-Weisfeiler conjecture, a weaker maximal-dimension statement: the maximal dimension of an irreducible module over any restricted Lie superalgebra should equal $p^{b_0/2}2^{\lfloor b_1/2\rfloor}$ , where $b_i$ is the maximum of $d_i(\chi)$ over all p-characters; he proved it for basic classical and completely solvable superalgebras. Ren, Shu, Yang and Zhang (2023) verified this maximal-dimension conjecture for both strange classical series and their derived algebras, and proved irreducibility of the relevant baby Verma modules (the analogues, for reduced enveloping superalgebras, of highest-weight modules: quotients of $U_\chi(\mathfrak g)$ induced from a character of a maximal isotropic subalgebra) for regular semisimple and regular nilpotent p-characters (regular meaning that the centralizer $\mathfrak g_\chi$ has the smallest possible dimension) on the periplectic series. The maximal-dimension statement, however, concerns one extremal p-character and does not yield divisibility for a prescribed p-character. The authors state explicitly that the Kac-Weisfeiler property itself seems difficult to judge for the strange classical series: its verification has neither been finished for the queer type nor been done at all for the periplectic type.

Problem Statement

Let $\mathbf k$ be an algebraically closed field of characteristic $p>2$ , and let $\mathfrak g$ be a strange classical Lie superalgebra over $\mathbf k$ , regarded as a restricted Lie superalgebra: the queer series $q(n)$ or the periplectic series $\tilde p(n)$ , together with their derived algebras $[q(n),q(n)]$ and $[\tilde p(n),\tilde p(n)]$ . For a p-character $\chi\in\mathfrak g_{\bar0}^*$ , extended to $\mathfrak g^*$ by $\chi(\mathfrak g_{\bar1})=0$ , let $\mathfrak g_\chi=\{y\in\mathfrak g\mid\chi([y,\mathfrak g])=0\}$ be its centralizer, and set $d_i=\dim\mathfrak g_i-\dim\mathfrak g_{\chi,i}$ for $i\in\{\bar0,\bar1\}$ . Question: does the Super Kac-Weisfeiler Property hold, that is, is the dimension of every irreducible module over the reduced enveloping superalgebra $U_\chi(\mathfrak g)$ divisible by $p^{d_0/2}\,2^{\lfloor d_1/2\rfloor}$ for every p-character $\chi$ ? The open cases are: (1) the queer series $q(n)$ with arbitrary, non-nilpotent p-characters, where by the reduction of Wang and Zhao the question is tied to their Morita super-equivalence conjecture relating $U_\chi(q(n))$ to $U_\chi(q(n)_{\chi_s})$ for the Jordan decomposition $\chi=\chi_s+\chi_n$ into semisimple and nilpotent parts; and (2) the periplectic series $\tilde p(n)$ and its derived algebra, with any p-character, for which no divisibility result is known.

References

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