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Let $G$ be a reductive algebraic group over an algebraically closed field ofpositive characteristic, $G_1$ the Frobenius kernel of $G$, and $T$ a maximaltorus of $G$. We show that the $G_1T$-Verma modules of singular highest weightsare all rigid, determine their Loewy length, and describe their Loewy structureusing the periodic Kazhdan-Lusztig $Q$-polynomials. We assume that thecharacteristic of the field is large enough that, in particular, Lusztig'sconjecture for the irreducible $G_1T$-characters hold.

Loewy structure of $G_1T$-Verma modules of singular highest weights