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

TL;DR

This paper investigates the structure of $G_1T$-Verma modules with singular highest weights in positive characteristic, establishing their rigidity, Loewy length, and structure via Kazhdan-Lusztig polynomials under certain assumptions.

Contribution

It proves the rigidity of these modules, determines their Loewy length, and describes their Loewy structure using Kazhdan-Lusztig $Q$-polynomials, assuming large enough characteristic.

Findings

All $G_1T$-Verma modules of singular highest weights are rigid.

The Loewy length of these modules is explicitly determined.

Their Loewy structure is described using periodic Kazhdan-Lusztig $Q$-polynomials.

Abstract

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 $G_1T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan-Lusztig $Q$-polynomials. We assume that the characteristic of the field is large enough that, in particular, Lusztig's conjecture for the irreducible $G_1T$-characters hold.