Super Kac–Weisfeiler divisibility for arbitrary restricted Lie superalgebras: irreducible module dimensions, the maximal-locus statement, and support-variety refinements

For an ordinary restricted Lie algebra in prime characteristic, the Kac–Weisfeiler theory predicts exactly how the dimensions of irreducible modules are constrained by the geometry of p-characters: Premet's theorem gives p-power divisibility by half the codimension of the centralizer of the p-character, Premet–Skryabin proved that the locus of p-characters carrying maximal-dimensional simples is a nonempty Zariski-open set, and the first Kac–Weisfeiler conjecture predicts the maximal dimension of all simple modules. For restricted Lie superalgebras the corresponding theory is only partially built: the fixed-character super divisibility (a mix of p-powers and 2-powers) and the maximal-dimension conjecture are proved for basic classical, strange classical, solvable, and gl-type superalgebras in characteristic p>2, but remain open for arbitrary restricted Lie superalgebras, the super analogue of the Premet–Skryabin openness statement is blocked by the failure of primeness of enveloping superalgebras, and the support-variety refinements are posed as open in odd characteristic. The problem is to settle these divisibility laws, the maximal-dimension formula, and the maximal-locus statement in their full generality over algebra, p-character, and prime characteristic.

Background

Over an algebraically closed field $k$ of characteristic $p>2$, a finite-dimensional Lie superalgebra $\mathfrak g=\mathfrak g_{\bar 0}\oplus\mathfrak g_{\bar 1}$ (a $\mathbb Z_2$-graded algebra with a super-commutative bracket) is called restricted if its even part carries a $p$-map $x\mapsto x^{[p]}$ on $\mathfrak g_{\bar 0}$ compatible with the bracket, making $\mathfrak g_{\bar 0}$ a restricted Lie algebra in the sense of Jacobson. On any irreducible $U(\mathfrak g)$-module, each central element $x^p-x^{[p]}$ ($x\in\mathfrak g_{\bar 0}$) acts by a scalar $\chi(x)^p$ for a linear functional $\chi\in\mathfrak g_{\bar 0}^*$ called the p-character; quotienting by the corresponding central ideal gives the reduced enveloping superalgebra $U_\chi(\mathfrak g)$, of dimension $p^{\dim\mathfrak g_{\bar 0}}\,2^{\dim\mathfrak g_{\bar 1}}$, and every irreducible $U(\mathfrak g)$-module lives in exactly one $U_\chi(\mathfrak g)$.

For ordinary restricted Lie algebras ($\mathfrak g_{\bar 1}=0$) Weisfeiler and Kac posed in 1971 two conjectures linking module dimensions to the orbit geometry of $p$-characters. Given $\chi$, the form $B_\chi(x,y)=\chi([x,y])$ is alternating and its radical $\mathfrak g_\chi=\{x:\chi([x,\mathfrak g])=0\}$ is the centralizer (stabilizer) of $\chi$; the codimension of $\mathfrak g_\chi$ is even. The second conjecture, proved by Premet for Lie algebras of reductive groups, states that every finite-dimensional $U_\chi(\mathfrak g)$-module has dimension divisible by $p^{(\dim\mathfrak g-\dim\mathfrak g_\chi)/2}$. Premet and Skryabin later developed a deformation framework and proved, for every restricted Lie algebra, that the set of $\chi$ for which all simple $U_\chi(\mathfrak g)$-modules attain the global maximal dimension $M(\mathfrak g)$ is nonempty and Zariski-open in $\mathfrak g^*$. The first conjecture predicts that this maximum equals $p^{(n-r)/2}$, where $n=\dim\mathfrak g$ and $r$ is the minimal centralizer dimension (the index); it remains open in general, proved only in large characteristics by Martin–Stewart–Topley (with a simplified proof by Tikaradze), and it fails without the restricted hypothesis (Topley's counterexample).

In the super setting the centralizer codimension splits as $d_0|d_1$ with $d_0=\dim\mathfrak g_{\bar 0}-\dim\mathfrak g_{\chi,\bar 0}$ even and $d_1=\dim\mathfrak g_{\bar 1}-\dim\mathfrak g_{\chi,\bar 1}$ possibly odd, and the odd part contributes powers of $2$ alongside the $p$-powers; when $d_1$ is odd, the extra factor of $2$ is supplied by simple modules 'of type Q', whose endomorphism ring is the two-dimensional Clifford algebra — a purely super phenomenon that breaks Schur-lemma arguments. Wang and Zhao formulated the Super Kac–Weisfeiler conjecture: every finite-dimensional $U_\chi(\mathfrak g)$-module has dimension divisible by $p^{d_0/2}\,2^{\lfloor d_1/2\rfloor}$, and proved it for basic classical Lie superalgebras (those with reductive even part and a nondegenerate even supersymmetric invariant form). Zhao's deformation approach reproduces the super analogue of the toral-centralizer case of Premet–Skryabin (the case where some p-character has a centralizer that is toral, i.e. spanned by elements $x$ with $x^{[p]}=x$), but the openness statement for the maximal locus does not transfer, because $U(\mathfrak g)$ need not be a prime ring for Lie superalgebras (Bell's counterexample), and primeness is essential in the original proof.

A second thread comes from support varieties: for a restricted Lie algebra the cohomology ring of the restricted enveloping algebra is finitely generated, and the support variety of a module (the maximal-ideal spectrum support of its self-Ext algebra, realizable as a rank variety inside $\mathfrak g$ by Friedlander–Parshall) has dimension tightly related to the $p$-divisibility of the module's dimension. Boe–Kujawa–Nakano built the char-zero super theory, and Drupieski–Kujawa established the characteristic-2 theory (rank varieties, i.e. detectable loci on which restricted elements act nilpotently, and the Balmer spectrum, the space of prime tensor-ideals of the stable module category) by exploiting that a Lie superalgebra in characteristic 2 has an underlying ordinary restricted Lie algebra; they state that only partial divisibility results are known in super representation theory, that improving them is an open direction, and that the odd-characteristic counterparts of their results remain open. Bin Shu recently unified the maximal-dimension side: for $\chi\in\mathfrak g_{\bar 0}^*$ let $\mathfrak z_\chi=\{X\in\mathfrak g:\chi([X,\mathfrak g])=0\}$ and $b_i^\chi=\dim\mathfrak g_{\bar i}-\dim\mathfrak z_{\chi,\bar i}$; his first super Kac–Weisfeiler conjecture predicts that the maximal dimension of irreducible $U(\mathfrak g)$-modules is $M(\mathfrak g)=\max_{\chi\in\mathfrak g_{\bar 0}^*}\,p^{b_0^\chi/2}\,2^{\lfloor b_1^\chi/2\rfloor}$, specializing to the ordinary first conjecture when $\mathfrak g_{\bar 1}=0$. It is verified for basic classical and completely solvable superalgebras, and for the strange classical (queer and periplectic) series, but not in the generality in which it is posed.

Problem Statement

Let $\mathfrak g=\mathfrak g_{\bar 0}\oplus\mathfrak g_{\bar 1}$ be an arbitrary finite-dimensional restricted Lie superalgebra over an algebraically closed field $k$ of characteristic $p>2$. For a $p$-character $\chi\in\mathfrak g_{\bar 0}^*$, let $\mathfrak z_\chi=\{X\in\mathfrak g:\chi([X,Y])=0\ \text{for all}\ Y\in\mathfrak g\}$ be its centralizer, and set $d_i(\chi)=\dim_k\mathfrak g_{\bar i}-\dim_k\mathfrak z_{\chi,\bar i}$ for $i\in\{0,1\}$; note $d_0(\chi)$ is even while $d_1(\chi)$ may be odd. Determine, in full generality over $\mathfrak g$, $\chi$, and $p>2$, the sharp divisibility laws for irreducible $\mathfrak g$-modules. Concretely, prove or refute, for every such $\mathfrak g$:

1. (Fixed-character super Kac–Weisfeiler divisibility, Wang–Zhao) Every finite-dimensional $U_\chi(\mathfrak g)$-module has dimension divisible by $p^{d_0(\chi)/2}\,2^{\lfloor d_1(\chi)/2\rfloor}$.

2. (First super Kac–Weisfeiler conjecture, Shu) The maximal dimension of an irreducible $U(\mathfrak g)$-module equals $M(\mathfrak g)=\max_{\chi\in\mathfrak g_{\bar 0}^*}\;p^{d_0(\chi)/2}\,2^{\lfloor d_1(\chi)/2\rfloor}.$ Here dimension means total $k$-dimension $\dim V_{\bar 0}+\dim V_{\bar 1}$ of the underlying vector space of the module, and the case $\mathfrak g_{\bar 1}=0$ recovers the ordinary first Kac–Weisfeiler conjecture, so a full answer in particular resolves that conjecture in characteristics $p>2$.

3. (Super Premet–Skryabin openness) Determine whether the set of $\chi\in\mathfrak g_{\bar 0}^*$ for which all simple $U_\chi(\mathfrak g)$-modules have dimension $M(\mathfrak g)$ is nonempty and Zariski-open in $\mathfrak g_{\bar 0}^*$, given that $U(\mathfrak g)$ need not be a prime ring and the original deformation argument therefore does not transfer.

These three items form one objective — the super Kac–Weisfeiler divisibility theory: (1) is the fixed-character core, (2) is (1) maximized over $p$-characters together with the assertion that the bound is attained, and (3) is the geometric regularity of the locus where attainment happens. As a further question, in the sense posed by Drupieski–Kujawa, relate the $p$- and $2$-adic divisibility of module dimensions to the dimensions of support varieties for restricted Lie superalgebras in odd characteristic; the characteristic-2 counterpart of the whole program under the $\mathbb Z_2$-graded convention, where the classical sign structure degenerates, is a further open direction.

References

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