Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

We show that the modules for the Frobenius kernel of a reductive algebraic group over an algebraically closed field of positive characteristic $p$ induced from the $p$-regular blocks of its parabolic subgroups can be $\mathbb{Z}$-graded. In particular, we obtain that the modules induced from the simple modules of $p$-regular highest weights are rigid and determine their Loewy series, assuming the Lusztig conjecture on the irreducible characters for the reductive algebraic groups, which is now a theorem for large $p$. We say that a module is rigid if and only if it admits a unique filtration of minimal length with each subquotient semisimple, in which case the filtration is called the Loewy series.