Complexity for modules over finite Chevalley groups and classical Lie algebras (Q1805882)

Let \(G\) be a connected reductive algebraic group defined over a finite (prime) field \(\mathbb{F}_p\) of \(p\) elements. In 1987, \textit{B. J. Parshall} [Proc. Symp. Pure Math. 47, 233-248 (1987; Zbl 0649.20043)] conjectured that for any \(G\)-module \(M\), projectivity of \(M\) when restricted to \(\text{Lie} (G)\) implies the projectivity of \(M\) when restricted to \(G(\mathbb{F}_p)\). In this paper the authors provide a comparison theorem relating the complexities of \(M\) over \(\text{Lie}(G)\) and over \(G(\mathbb{F}_p)\) and Parshall's conjecture is proved as a direct consequence. The converse to Parshall's conjecture is also discussed. The answer is affirmative if the weights of \(M\) are not too large and negative in general.