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 ${\mathrm{St}}_r\otimes L(\lambda )$ is a tilting module for $L(\lambda )$ an irrreducible representation of $p^r$-restricted highest weight, and (iv) whether ${\mathrm{Ext}}_{G_1}^1(L(\lambda ),L(\mu ){)}^{(-1)}$ 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 $p\ge 2h-4$ 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.