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 St_r⊗L(λ) is a tilting module for L(λ) an irreducible representation of p^r-restricted highest weight, and (iv) whether Ext_G1¹(L(λ),L(μ))⁽⁻¹⁾ is a tilting module where L(λ) and L(μ) have p-restricted highest weight. The authors establish affirmative answers to each of these questions with a new uniform bound, namely p≥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.