Monodromy of Galois representations and equal-rank subalgebra equivalence

Details Monodromy of Galois representations and equal-rank subalgebra equivalence Monodromy of Galois representations and equal-rank subalgebra equivalence

2012-04-24

arxiv.org/abs/1204.5271v1

Mathematics

Number Theory

Scientific paper

Let K be a number field, P the set of prime numbers, and {\rho_l}_{l\in P} a compatible system (in the sense of Serre [17]) of semisimple, n-dimensional l-adic representations of Gal(\bar{K}/K). Denote the Zariski closure of \rho_l(Gal(\bar{K}/K)) in GL_{n,Q_l} by G_l and its Lie algebra by g_l. We know that G_l^\circ (the connected component) is reductive and the formal character of the tautological representation G_l^\circ -> GL_{n,Q_l} 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^\circ)^der -> GL_{n,Q_l} is likewise independent of l. By investigating the geometry of weights of this representation, we prove that the semisimple parts of g_l\otimes C satisfy an equal-rank subalgebra equivalence for all l. In particular, the number of A_n:=sl_{n+1,C} factors for n\in {6,9,10,11,...} and the parity of the number of A_4 factors in g_l\otimes C are independent of l.

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