The second Kac-Weisfeiler divisibility conjecture for truncated current Lie algebras

For a standard reductive group $G$ over an algebraically closed field of characteristic $p>0$ , the truncated current Lie algebra $\mathfrak{g}_m=\mathfrak{g}[t]/(t^{m+1})$ is a restricted, non-reductive Lie algebra that shares many structural features with the reductive Lie algebra $\mathfrak{g}=\operatorname{Lie}(G)$ . For a $p$ -character $\chi\in(\mathfrak{g}_m)^*$ , the second Kac-Weisfeiler conjecture (KW2) for $\mathfrak{g}_m$ predicts that $p^{\frac{1}{2}(\dim\mathfrak{g}_m-\dim\mathfrak{g}_m^{\chi})}$ divides the dimension of every simple module over the reduced enveloping algebra $U_\chi(\mathfrak{g}_m)$ , where $\mathfrak{g}_m^{\chi}$ is the stabilizer of $\chi$ . KW2 is a theorem for reductive $\mathfrak{g}$ (the case $m=0$ , due to Premet), and Chaffe and Topley recently established it for homogeneous, semisimple and regular $p$ -characters of $\mathfrak{g}_m$ , and for all $p$ -characters of $(\mathfrak{sl}_2)_m$ when $p>2$ . The problem asks whether KW2 holds for arbitrary $p$ -characters of $\mathfrak{g}_m$ for every standard reductive $\mathfrak{g}$ and $m\ge 1$ ; this general case is open.

Background

Over an algebraically closed field $\Bbbk$ of characteristic $p>0$ , the Lie algebra $\mathfrak{g}$ of an algebraic group carries a natural $p$ -power map $x\mapsto x^{[p]}$ compatible with the adjoint action, making it a restricted Lie algebra (also called a Lie $p$ -algebra). In the universal enveloping algebra $U(\mathfrak{g})$ the elements $x^{p}-x^{[p]}$ are central and generate the $p$ -centre $Z_p(\mathfrak{g})$ , so every simple $\mathfrak{g}$ -module is killed by relations $x^{p}-x^{[p]}=\chi(x)^{p}$ for a linear functional $\chi\in\mathfrak{g}^{*}$ , its $p$ -character. The quotient $U_\chi(\mathfrak{g})$ is the reduced enveloping algebra, a finite-dimensional algebra of dimension $p^{\dim\mathfrak{g}}$ whose simple modules are exactly the simple $\mathfrak{g}$ -modules with character $\chi$ . Write $\mathfrak{g}^{\chi}=\{x\in\mathfrak{g}:\chi([x,\mathfrak{g}])=0\}$ for the stabilizer of $\chi$ ; note that $\dim\mathfrak{g}-\dim\mathfrak{g}^{\chi}$ is the dimension of the coadjoint orbit of $\chi$ .

Kac and Weisfeiler (1971) conjectured that the representation theory of $U_\chi(\mathfrak{g})$ mirrors the geometry of the coadjoint orbit of $\chi$ , in the spirit of Kirillov's orbit method. Two of their statements became classical: KW1, that the maximal dimension of a simple $U(\mathfrak{g})$ -module equals $p^{\frac{1}{2}(\dim\mathfrak{g}-\operatorname{ind}\mathfrak{g})}$ , where $\operatorname{ind}\mathfrak{g}$ is the index of $\mathfrak{g}$ (the minimal dimension of a stabilizer $\mathfrak{g}^{\chi}$ ); and KW2, that for $\mathfrak{g}$ the Lie algebra of a reductive group, $p^{\frac{1}{2}(\dim\mathfrak{g}-\dim\mathfrak{g}^{\chi})}$ divides the dimension of every simple $U_\chi(\mathfrak{g})$ -module. Premet proved KW2 for reductive $\mathfrak{g}$ in 1995, and in his Mathematical Reviews review of Premet's paper Kac conjectured that KW2 should hold for the Lie algebra of every algebraic group; outside the reductive and the completely solvable cases, almost nothing is known about this general conjecture.

The problem studied here lives in the first structurally rich test case beyond the reductive world: truncated current Lie algebras. For $m\ge 1$ , $\mathfrak{g}_{m}=\mathfrak{g}\otimes_{\Bbbk}\Bbbk[t]/(t^{m+1})$ is the Lie algebra of the $m$ -th jet group scheme $J_mG$ (also called a Takiff algebra, especially for $m=1$ ); its bracket is taken componentwise in the $t$ -grading, and its $[p]$ -map sends $x\otimes t^{i}$ to $x^{[p]}\otimes t^{ip}$ , which is zero once $ip>m$ , so $\mathfrak{g}_m$ is very far from reductive. Friedlander and Parshall created the main tools for studying $U_\chi(\mathfrak{g})$ when $\mathfrak{g}$ is reductive: support varieties (affine varieties attached to modules and to $p$ -characters via the cohomology of reduced enveloping algebras, which measure the size of a module), the reduction of the module category by Morita equivalence (an equivalence of module categories) to nilpotent $p$ -characters, and the important class of nilpotent elements of standard Levi type (regular nilpotent elements of Levi subalgebras).

Chaffe and Topley (2023/2024) developed the modular representation theory of $\mathfrak{g}_m$ when $G$ is a standard reductive group (a reductive group satisfying the standard mild hypotheses on $G$ and the characteristic, reviewed in their paper, under which nilpotent orbits and the invariant bilinear form behave classically). Using the $G_m$ -invariant bilinear form on $\mathfrak{g}_m$ obtained by pairing equal $t$ -degrees via the invariant form $\kappa$ of $\mathfrak{g}$ , they identify $\mathfrak{g}_m$ with its dual and write every $p$ -character as $\chi=\hat\kappa_m(x)$ for some $x\in\mathfrak{g}_m$ ; the character is called semisimple or nilpotent when $x$ can be chosen so, homogeneous of degree $k$ when $\chi$ kills all $t$ -degrees except $t^{k}$ , and regular when $\dim\mathfrak{g}_m^{\chi}$ is minimal, equal to $\operatorname{ind}\mathfrak{g}_m$ . They prove a Morita equivalence reducing the study of $U_\chi(\mathfrak{g}_m)$ -modules to nilpotent $p$ -characters, analogous to the Friedlander-Parshall reduction, and they settle KW1 for $\mathfrak{g}_m$ completely. For KW2 their key tool is Premet's criterion from the theory of finite $W$ -algebras (associative algebras obtained by quantizing slices transverse to nilpotent orbits in $\mathfrak{g}$ ): if $\mathfrak{g}_m$ contains a $\chi$ -admissible subalgebra $\mathfrak{n}$ (a nilpotent subalgebra with $\chi$ vanishing on the $[p]$ -closure of $[\mathfrak{n},\mathfrak{n}]$ and meeting the support variety of $\chi$ trivially) with $\dim\mathfrak{n}=\frac{1}{2}(\dim\mathfrak{g}_m-\dim\mathfrak{g}_m^{\chi})$ , then KW2 holds for $\chi$ .

With this criterion, Chaffe and Topley prove that KW2 holds for $\mathfrak{g}_m$ whenever $\chi$ is homogeneous, semisimple or regular, and they verify KW2 for all $p$ -characters of $(\mathfrak{sl}_2)_m$ when $p>2$ . However, as they state explicitly, they do not show the conjecture for all $p$ -characters. For a mixed nilpotent character, such as $\chi=\hat\kappa_m(x)$ with $x$ combining different $t$ -degrees and non-regular nilpotent leading term, no admissible subalgebra of the required dimension is known; moreover the technique of baby Verma modules (the positive-characteristic analogue of induced highest-weight modules, built by inducing a one-dimensional module of a Borel-type subalgebra on whose nilradical $\chi$ vanishes) fails for such characters, since there may be no Borel-type subalgebra $\mathfrak{b}_m^{+}\subset\mathfrak{g}_m$ whose nilradical is killed by $\chi$ (already for $\mathfrak{g}=\mathfrak{sl}_2$ , $m=1$ , $\chi=\hat\kappa_m(e+ft)$ ).

Problem Statement

Let $\Bbbk$ be an algebraically closed field of characteristic $p>0$ , let $G$ be a standard reductive group over $\Bbbk$ (a reductive group satisfying the standard hypotheses on the group and the characteristic, as in the source paper, under which nilpotent orbits and the invariant bilinear form behave as in characteristic zero) with $\mathfrak{g}=\operatorname{Lie}(G)$ , let $m\ge 1$ , and let $\mathfrak{g}_m=\mathfrak{g}\otimes_{\Bbbk}\Bbbk[t]/(t^{m+1})$ be the $m$ -th truncated current Lie algebra, with componentwise bracket and restricted structure given by $(x\otimes t^{i})^{[p]}=x^{[p]}\otimes t^{ip}$ . For a $p$ -character $\chi\in(\mathfrak{g}_m)^{*}$ , let $\mathfrak{g}_m^{\chi}=\{x\in\mathfrak{g}_m:\chi([x,\mathfrak{g}_m])=0\}$ be the stabilizer, and let $U_\chi(\mathfrak{g}_m)$ be the reduced enveloping algebra. The problem is the second Kac-Weisfeiler divisibility conjecture for truncated current Lie algebras, the truncated-current instance of Kac's conjecture that KW2 holds for the Lie algebra of every algebraic group: is it true, for every standard reductive $G$ , every $m\ge 1$ , every $p$ -character $\chi$ of $\mathfrak{g}_m$ , and every simple $U_\chi(\mathfrak{g}_m)$ -module $M$ , that $p^{\frac{1}{2}(\dim\mathfrak{g}_m-\dim\mathfrak{g}_m^{\chi})}$ divides $\dim M$ ? The cases where $\chi$ is homogeneous, semisimple or regular, and the case $\mathfrak{g}=\mathfrak{sl}_2$ with $p>2$ (all $\chi$ ), are already settled affirmatively, and the Morita reductions of the source paper indicate that the main remaining difficulty lies in nilpotent $p$ -characters that are neither homogeneous nor regular; the question is asked at full generality, with no restriction on $\chi$ , $m$ , $\mathfrak{g}$ or $p$ beyond the standing standard-reductivity hypotheses.

References

Matthew Chaffe and Lewis Topley, Modular Representations of Truncated current Lie algebras, arXiv:2311.08208 (v1 2023, v2 2024), DOI 10.48550/arXiv.2311.08208, https://arxiv.org/abs/2311.08208
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