Monodromy of Galois representations and equal-rank subalgebra equivalence

Abstract

Let K be a number field, P the set of prime numbers, and {ρ`}`∈P a compatible system (in the sense of Serre [19]) of semisimple, n-dimensional `-adic representations of Gal(K/K). Denote the Zariski closure of ρ`(Gal(K/K)) in GLn,Q` by G` and its Lie algebra by g`. It is known that the identity component G ◦ ` is reductive and the formal character of the tautological representation G` ↪→ GLn,Q` is independent of ` (Serre). We use the theory of abelian `-adic representations to prove that the formal character of the tautological representation of the derived group (G` ) der ↪→ GLn,Q` is likewise independent of `. By investigating the geometry of weights of this faithful representation, we prove that the semisimple parts of g` ⊗ C satisfy an equal-rank subalgebra equivalence for all ` which is equivalent to the number of An := sln+1,C factors for n ∈ {6, 9, 10, 11, ...} and the parity of the number of A4 factors in g` ⊗ C are independent of `.