Monodromy of galois representations and equal-rank subalgebra equivalence

Abstract

Let K be a number field, P the set of prime numbers, and {ρℓ}ℓepsiv;P a compatible system (in the sense of Serre [19]) of semisimple, n-dimensional ℓ-adic representations of Gal(K/K). Denote the Zariski closure of ρ ℓ(Gal(K/K)) in GLn,ℚℓ by Gℓ and its Lie algebra by gℓ. It is known that the identity component G° ℓ is reductive and the formal character of the tautological representation G° ℓ ←hk; GLn,ℚℓ is independent of ℓ (Serre). We use the theory of abelian ℓ-adic representations to prove that the formal character of the tautological representation of the derived group (G° ℓ)der ←hk; GLn,ℚℓ is likewise independent of ℓ. By investigating the geometry of weights of this faithful representation, we prove that the semisimple parts of gℓ⊗ c satisfy an equal-rank subalgebra equivalence for all ℓ which is equivalent to the number of An := sln+1,c factors for n epsiv; {6, 9, 10, 11, . . .} and the parity of the number of A4 factors in gℓ⊗ c are independent of ℓ. © International Press 2013.