On the Humphreys-Verma Conjecture for semisimple algebraic groups of rank $2$

Let $G$ be a connected, semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic. For each restricted dominant weight $\lambda$, there is the associated principal indecomposable $G_1$-module $Q_1(\lambda )$, where $G_1$ is the first infinitesimal subgroup of $G$. The assertion that, for every such $\lambda$, there exists a $G$-module whose restriction to $G_1$ is isomorphic to $Q_1(\lambda )$ is known as the Humphreys–Verma Conjecture. For groups of rank $2$, it was shown in \cite{BNPS1} that the Humphreys–Verma Conjecture holds in all cases except one, namely when $G$ is of type $G_2$, the characteristic is $2$, and $\lambda =0$. This case remained completely open. Moreover, in every previously resolved case, the module $Q_1(\lambda )$ could be realized as the restriction of a suitable tilting module. However, in \cite{BNPS2} it was shown that $Q_1(0)$ for $G_2$ in characteristic $2$ cannot arise as the restriction of a tilting module, thereby providing the first counterexample to a conjecture of the first author. In this paper, we construct a $G$-module whose restriction to $G_1$ is $Q_1(0)$, thereby establishing the Humphreys–Verma Conjecture in the last remaining rank $2$ case. Our construction provides the first known example of a $G$-structure on a principal indecomposable $G_1$-module that does not arise from a tilting module. This reveals a new phenomenon in the study of the Humphreys–Verma Conjecture and suggests new directions for understanding $G$-structures on principal indecomposable $G_1$-modules.