Gradings by cyclic groups on classical simple Lie algebras in prime characteristics

Abstract

We classify, up to isomorphism, gradings by finite cyclic groups on classical simple Lie algebras $\mathfrak{g}$ over an algebraically closed field of arbitrary characteristic. Using the smoothness of the automorphism group scheme $\operatorname{\mathbf{Aut}}\mathfrak{g}$ and correspondence between $\mathbb Z_m$-gradings and morphisms $\boldsymbol{\mu}_m\to\operatorname{\mathbf{Aut}}\mathfrak{g}$, we express the classification as an orbit problem for certain Weyl-type groups. More generally, for the affine group scheme $\mathbf G$ associated to a semisimple algebraic group $G$ and the constant group scheme $\mathbf \Gamma_0$ associated to a subgroup $\Gamma_0$ of the automorphism group of the based root datum of $G$, we consider the classification of morphisms $\boldsymbol{\mu}_m\to \mathbf G \rtimes \mathbf\Gamma_0$ up to conjugation by $G \rtimes \Gamma_0$. We show that the classification in characteristic $p$ is the same as in characteristic $0$ except that, in characteristic $p$, only elements of $\Gamma_0$ whose order is prime to $p$ can occur. For $\mathbb{Z}_m$-gradings on $\mathfrak{g}$, this extends the classification by Kac coordinates to arbitrary characteristic, with the caveat that only diagram automorphisms of order prime to $p$ are allowed and, if $p=2$ or $3$, the type of $\operatorname{Aut}\mathfrak{g}$ is not always the same as the type of $\mathfrak{g}$.