16.39 (2006)

(J. E. Humphreys, D. N. Verma). Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. Let $\mathfrak{g}$ be the Lie algebra of $G$ and let $u = u(\mathfrak{g})$ be the restricted enveloping algebra of $\mathfrak{g}$. By a theorem of Curtis every irreducible restricted $u$-module (i.e. every irreducible restricted $\mathfrak{g}$-module) is the restriction to $\mathfrak{g}$ of a (rational) $G$-module. Is it also true that every projective indecomposable $u$-module is the restriction of a rational $G$-module? This is true if $p \geqslant 2h - 2$ (where $h$ is the Coxeter number of $G$) by results of Jantzen.