Monodromy of Galois representations and equal-rank subalgebra equivalence

Let $K$ be a number field, $\mathscr{P}$ the set of prime numbers, and $\{\rho_\ell\}_{\ell\in \mathscr{P}}$ a compatible system (in the sense of Serre) of semisimple, $n$-dimensional $\ell$-adic representations of $\mathrm{Gal}(\overline{K}/K)$. Denote the Zariski closure of $\rho_\ell(\mathrm{Gal}(\overline{K}/K))$ in $\mathrm{GL}_{n,\mathbb{Q}_\ell}$ by $G_\ell$ and its Lie algebra by $\mathfrak{g}_\ell$. It is known that the identity component $G_\ell^\circ$ is reductive and the formal character of the tautological representation $G_\ell^\circ\hookrightarrow \mathrm{GL}_{n,\mathbb{Q}_\ell}$ is independent of $\ell$ (Serre). We use the theory of abelian $\ell$-adic representations to prove that the formal character of the tautological representation of the derived group $(G_\ell^\circ)^{{\rm der}}\hookrightarrow \mathrm{GL}_{n,\mathbb{Q}_\ell}$ is likewise independent of $\ell$. By investigating the geometry of weights of this faithful representation, we prove that the semisimple parts of $\mathfrak{g}_\ell\otimes\mathbb{C}$ satisfy an equal-rank subalgebra equivalence for all $\ell$, which is equivalent to the number of $A_n:=\mathfrak{sl}_{n+1,\mathbb{C}}$ factors for $n\in\{6,9,10,11,\ldots\}$ and the parity of the number of $A_4$ factors in $\mathfrak{g}_\ell\otimes\mathbb{C}$ are independent of $\ell$.

2010 Mathematics Subject Classification

11F03, 11F11

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Published 13 March 2014