On the simplicity of induced modules for reductive Lie algebras. II (Q1621596)

Let $\mathfrak{g}$ be the Lie algebra of a reductive algebraic group $G$ that is defined over an algebraically closed field $\mathbf{F}$ of prime characteristic $p$. The main result of the paper under review is a sufficient condition for the simplicity of a $\mathfrak{g}$-module that is induced from a simple module over a parabolic subalgebra $\mathfrak{p}_I$ of $\mathfrak{g}$. Such a $\mathfrak{g}$-module is simple provided a certain polynomial function on the linear dual of a common Cartan subalgebra of $\mathfrak{g}$ and $\mathfrak{p}_I$ does not vanish on the highest weight of the simple $\mathfrak{p}_I$-module. In a previous paper the author had to assume that the $p$-character of the induced module vanishes on the unipotent radical of the opposite parabolic subalgebra of $\mathfrak{p}_I$. Under this hypothesis the condition on the highest weight is also necessary. In the present paper it is shown that the sufficiency of the condition remains valid without this restriction on the $p$-character. So the main result of the paper under review answers one half of a question of \textit{E. M. Friedlander} and \textit{B. J. Parshall} [Am. J. Math. 112, No. 3, 375--395 (1990; Zbl 0714.17007)] under some mild assumptions on $G$, $\mathfrak{g}$, and $p$. In a preprint [``A criterion for (the) irreducibility of parabolic baby Verma modules of reductive Lie algebras'', Preprint, \url{arXiv.1404.4945}] \textit{Y.-Y. Li} et al. gave another sufficient condition for the simplicity of a $\mathfrak{g}$-module induced from a simple module over certain parabolic subalgebras $\mathfrak{p}$ for $\mathfrak{g}=\mathfrak{sl}_n(\mathbf{F})$, $\mathfrak{so}_n (\mathbf{F})$, or $\mathfrak{sp}_{2n}(\mathbf{F})$ when the $p$-character is of standard Levi form and the high weight of the simple $\mathfrak{p}$-module is $p$-regular. At the end of his paper the author shows that for a $p$-character of standard Levi form his simplicity criterion only applies to irreducible $\mathfrak{p}$-modules whose high weight is not $p$-regular. As a consequence, the author's simplicity criterion complements the results of Li, Shu, and Yao.