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Suppose $g=g_0+g_1$ is a finite-dimensional restricted Lie superalgebra overan algebraically closed field $k$ of characteristic $p>2$. In this article, wepropose a conjecture for maximal dimensions of irreducible modules over theuniversal enveloping algebra $U(g)$ of $g$, as a super generalization of thecelebrated first Kac-Weisfeiler conjecture. It is demonstrated that theconjecture holds for all basic classical Lie superalgebras and all completelysolvable restricted Lie superalgebras. In this process, we investigateirreducible representations of solvable Lie superalgebras.

Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture