Suppose $\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf {k}$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$, as a super generalization of the celebrated first Kac–Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras.

假设$\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ 是代数闭域$\mathbf {k上的有限维受限李超代数}$特征$p>2$。在本文中，我们提出了关于 $\mathfrak {g}$ 的通用包络代数 $U(\mathfrak {g})$ 上不可约模最大维数的猜想，作为著名的第一个 Kac-Weisfeiler 猜想的超概括。证明了该猜想对于所有基本经典李超代数和所有完全可解的限制李超代数都成立。在此过程中，我们研究可解李超代数的不可约表示。