Tikaradze's Kac–Weisfeiler-type conjecture for filtered algebras in positive characteristic: $i(\chi)\ge\tfrac{1}{2}s(\chi)$

For an algebra $A$ finite over its center over an algebraically closed field of characteristic $p>2$, one asks how the $p$-power divisibility of the dimensions of simple modules with central character $\chi$ is controlled by Poisson geometry. Tikaradze (J. Algebra 383, 2013) proved a Kac–Weisfeiler-type divisibility theorem for a class of filtered algebras including rational Cherednik algebras, but posed as Conjecture 1 a general analogue of the Kac–Weisfeiler conjecture that remains unresolved: if $A$ is nonnegatively filtered with $\operatorname{gr}A$ a finitely generated commutative domain such that $(\operatorname{gr}A)^p\subset\operatorname{gr}\mathbf{Z}(A)$ and $\operatorname{Spec}\operatorname{gr}A$ is a Cohen–Macaulay variety that is a union of finitely many symplectic leaves, then the $p$-exponent $i(\chi)$ of every central character should satisfy $i(\chi)\ge\tfrac{1}{2}s(\chi)$, where $s(\chi)$ is the dimension of the smooth stratum through $\chi$. The problem is to prove or refute this conjecture for the full class of algebras and all central characters, including singular ones.

Background

Let $k$ be an algebraically closed field of prime characteristic $p>2$. For a $k$-algebra $A$ that is finitely generated as a module over its center $\mathbf{Z}(A)$ (equivalently in this setting, a PI algebra, i.e. an algebra satisfying a nontrivial polynomial identity), every simple $A$-module is finite-dimensional and is annihilated by a unique maximal ideal of $\mathbf{Z}(A)$; a central character is a $k$-point $\chi\in\operatorname{Spec}\mathbf{Z}(A)$, and one writes $i(\chi)$ for the largest integer such that $p^{i(\chi)}$ divides the dimension of every simple $A$-module with central character $\chi$. The guiding question of modular (positive-characteristic) representation theory is to bound these $p$-powers by geometry.

The prototype is the Kac–Weisfeiler conjecture (Kac–Weisfeiler, 1976), proved by Premet (1995): if $\mathfrak{g}$ is the Lie algebra of a semisimple simply-connected algebraic group over $k$ and $\chi\in\mathfrak{g}^*$, then every irreducible representation of the restricted enveloping algebra $U_\chi(\mathfrak{g})$ (the quotient of the universal enveloping algebra by the ideal generated by $x^p-x^{[p]}-\chi(x)$ for $x\in\mathfrak{g}$, where $x\mapsto x^{[p]}$ is the $p$-power map of the restricted Lie algebra structure) has dimension divisible by $p^{\frac{1}{2}\dim G\cdot\chi}$, where $G\cdot\chi$ is the coadjoint orbit, a symplectic leaf of $\mathfrak{g}^*$. Premet and Skryabin (1999) recast and generalized the dimension-divisibility mechanism for restricted Lie algebras.

Tikaradze (2013) transferred this question to arbitrary filtered algebras. Suppose $A$ carries a nonnegative increasing filtration such that the associated graded algebra $\operatorname{gr}A$ (the algebra of successive filtration quotients) is a finitely generated commutative domain. Then $\operatorname{gr}A$ carries a natural Poisson bracket (a bilinear antisymmetric derivation rule $\{f,g\}$ induced from the commutator in $A$), making $\operatorname{Spec}\operatorname{gr}A$ a Poisson variety. A symplectic leaf of a Poisson variety is a maximal locally closed smooth subvariety on which the Poisson bracket restricts to a nondegenerate (symplectic) form; a Poisson variety is a union of such leaves, and here one assumes the union is finite. Two further hypotheses echo modular phenomena: the Frobenius-type condition $(\operatorname{gr}A)^p\subset\operatorname{gr}\mathbf{Z}(A)$ (all $p$-th powers in the associated graded are central, the analogue of the $p$-center of the enveloping algebra), and that $\operatorname{Spec}\operatorname{gr}A$ is a Cohen–Macaulay variety (its coordinate ring has depth equal to its dimension, a standard homological regularity condition on singularities). Finally, $\operatorname{Spec}\mathbf{Z}(A)$ admits a stratification by smooth strata — in the source's intended reading these strata correspond, via the deformation from $\operatorname{gr}A$ to $A$ and the Frobenius twist (the base change of a variety along the map $x\mapsto x^p$ on its coordinate ring), to the symplectic leaves of $\operatorname{Spec}\operatorname{gr}A$ — and $s(\chi)$ denotes the dimension of the smooth stratum containing $\chi$. This framework, that of Poisson orders introduced by Brown–Gordon (2003), covers restricted enveloping algebras of semisimple Lie algebras (where $\operatorname{gr}A$ is the coordinate ring of the nilpotent cone (the variety of nilpotent elements), by work of Veldkamp), symplectic reflection and rational Cherednik algebras in characteristic $p$ (Etingof–Ginzburg, 2002; Brown–Changtong, 2010), and many algebras of differential-operator type. For these algebras the irreducible modules are governed, on the Azumaya locus (the largest open subset of $\operatorname{Spec}\mathbf{Z}(A)$ over which the algebra is an Azumaya algebra, i.e. locally a matrix algebra over its center), by the crystalline differential operators of Bezrukavnikov–Mirković–Rumynin (2008), a characteristic-$p$ analogue of the sheaf of differential operators that forms an Azumaya sheaf on the Frobenius twist of the cotangent bundle.

For the enveloping-algebra specialization the conjecture below reduces exactly to Premet's theorem, since there $i(\chi)=\tfrac{1}{2}\dim G\cdot\chi$. Tikaradze proved the inequality on the smooth locus of $\operatorname{Spec}\mathbf{Z}(A)$ (where $A$ is Azumaya over its center, by his 2011 result on the Azumaya locus of almost commutative algebras), established general lower bounds of Kac–Weisfeiler type via a deformation to the associated graded and Poisson-module divisibility (following Premet–Skryabin), and established the expected Kac–Weisfeiler-type divisibility — which the paper itself notes is weaker than its Conjecture 1 — for rational Cherednik algebras and their spherical subalgebras (the invariant parts $eH_ce$ for the symmetrizing idempotent $e$), using the characteristic-$p$ version of the Bezrukavnikov–Etingof parabolic induction/restriction decomposition (2009) to reduce local studies to stabilizer subgroups. However, at singular central characters — outside the Azumaya locus — no argument is known that produces the expected $p^{\frac{1}{2}s(\chi)}$ divisibility for a general algebra in the class, and this is precisely the content of the open conjecture.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $p>2$, and let $A$ be an affine $k$-algebra that is finite as a module over its center $\mathbf{Z}(A)$ (under this standing hypothesis of the source, every simple $A$-module is finite-dimensional and is annihilated by a unique maximal ideal of $\mathbf{Z}(A)$). Suppose that $A$ is nonnegatively filtered and that:

1. $\operatorname{gr}A$ is a finitely generated commutative domain over $k$;
2. the Frobenius powers satisfy $(\operatorname{gr}A)^p\subset\operatorname{gr}\mathbf{Z}(A)$;
3. $\operatorname{Spec}\operatorname{gr}A$ is a union of finitely many symplectic leaves and is a Cohen–Macaulay variety.

For a central character $\chi\in\operatorname{Spec}\mathbf{Z}(A)$ (a $k$-point, i.e. a maximal ideal of $\mathbf{Z}(A)$), let $i(\chi)$ be the largest integer such that $p^{i(\chi)}$ divides the $k$-dimension of every simple $A$-module with central character $\chi$, and let $s(\chi)$ be the dimension of the smooth stratum containing $\chi$ in the smooth stratification of $\operatorname{Spec}\mathbf{Z}(A)$ (in the source's reading these strata correspond, via the deformation to the associated graded and the Frobenius twist, to the symplectic leaves of $\operatorname{Spec}\operatorname{gr}A$).

This is Conjecture 1 of Tikaradze's 'An analogue of the Kac–Weisfeiler conjecture' (J. Algebra 383 (2013) 168–177), generalizing Premet's theorem on the Kac–Weisfeiler conjecture from restricted enveloping algebras to arbitrary filtered algebras of the above class. Prove, or refute by an explicit counterexample, that for every such algebra $A$ and every central character $\chi$,

$$i(\chi)\;\ge\;\tfrac{1}{2}\,s(\chi).$$

References

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