Let $F$ be a CM field. Given a regular algebraic self-dual cuspidal automorphic representation $\Pi$ of $GL_n(\mathbb A_F)$ and a prime $l$, there is a continuous Galois representation $r_{\Pi}: \Gamma_F \to GL_n(\overline{\mathbb Q}_l)$ attached to $\Pi$ satisfying a number of properties, thanks to the work of many people. See, for example, Theorem 1.2 of Sug Woo Shin's work: https://math.berkeley.edu/~swshin/StableGal.pdf. In particular, the Galois character $\det r_{\Pi}$ corresponds to under class field theory the central character of $\Pi$.
It follows from the above and the work of Arthur that to a cuspidal automorphic representation $\pi$ of $Sp_{2n}(\mathbb A_F)$ we can attach Galois representations $r_{\pi}: \Gamma_F \to SO_{2n+1}(\overline{\mathbb Q}_l)$, see Theorem 2.4 of https://arxiv.org/pdf/1609.04223.pdf. My question is, can we extract information on the central character of $\pi$ from the Galois side just like in the $GL_n$ case?
