Loewy series of parabolically induced $G_1T$-Verma modules

TL;DR

This paper proves that certain modules induced from simple modules in positive characteristic are rigid and explicitly determines their Loewy series, assuming the Lusztig conjecture, now a theorem for large p.

Contribution

It establishes the rigidity and Loewy series of parabolically induced modules for Frobenius kernels under the Lusztig conjecture, extending understanding of module structure in positive characteristic.

Findings

Modules are rigid under the given assumptions.

Loewy series of these modules are explicitly determined.

Results hold for large prime characteristic p.

Abstract

Assuming the Lusztig conjecture on the irreducible characters for reductive algebraic groups in positive characteristic $p$, which is now a theorem for large $p$, we show that the modules for their Frobenius kernels induced from the simple modules of $p$-regular highest weights for their parabolic subgroups are rigid and determine their Loewy series.