Monodromy of Galois representations and equal-rank subalgebra equivalence

TL;DR

This paper proves that certain structural features of monodromy groups associated with compatible systems of l-adic representations are independent of l, revealing deep uniformities in their algebraic properties across different primes.

Contribution

It establishes l-independence of the formal character and an equal-rank subalgebra equivalence for the Lie algebras of monodromy groups in compatible systems.

Findings

Formal character of the derived group is independent of l

Equal-rank subalgebra equivalence holds for all l

Number of A_n factors and parity of A_4 factors are l-independent

Abstract

We study l-independence of monodromy groups G_l of any compatible system of l-adic representations (in the sense of Serre) of number field K assuming semisimplicity. We prove that the formal character of the derived group of the identity component of G_l is independent of l and the (complexified) Lie algebra g_l of G_l satisfies an equal-rank subalgebra equivalence for all l. This equivalence is equivalent to the l-independence of the number of A_n factors for all n belonging to {6,9,10,11,...} and the parity of A_4 factors in g_l.