On Donkin's Tilting Module Conjecture III: New Generic Lower Bounds

In this paper the authors consider four questions of primary interest for the representation theory of reductive algebraic groups: (i) Donkin's Tilting Module Conjecture, (ii) the Humphreys-Verma Question, (iii) whether $\operatorname{St}_r \otimes L(\lambda)$ is a tilting module for $L(\lambda)$ an irrreducible representation of $p^{r}$-restricted highest weight, and (iv) whether $\operatorname{Ext}^{1}_{G_{1}}(L(\lambda),L(\mu))^{(-1)}$ is a tilting module where $L(\lambda)$ and $L(\mu)$ have $p$-restricted highest weight. The authors establish affirmative answers to each of these questions with a new uniform bound, namely $p\geq 2h-4$ where $h$ is the Coxeter number. Notably, this verifies these statements for infinitely many more cases. Later in the paper, questions (i)-(iv) are considered for rank two groups where there are counterexamples (for small primes) to these questions.