Rectangular representations and $λ$-independence of algebraic monodromy groups

Abstract

Let $\mathfrak g$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak g$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $λ$-independence results on the algebraic monodromy groups of compatible systems of $λ$-adic Galois representations of number fields.