Let $\Gamma$ be a topological group, $n \geq 1$ an integer, $\ell$ a prime number, and $\overline{\mathbb{Q}}_{\ell}$ the algebraic closure of the $\ell$-adic integers. We set $\Phi(GSpin_{2n + 1})$ to be the set of continuous maps $\phi: \Gamma \rightarrow GSpin_{2n + 1}(\overline{\mathbb{Q}}_{\ell})$ valued in the general spin group up to $GSpin_{2n + 1}(\overline{\mathbb{Q}}_{\ell})$-conjugacy and $\Phi(GL_{2^{n}})$ to be the set of continuous maps $\phi: \Gamma \rightarrow GL_{2^{n}}(\overline{\mathbb{Q}}_{\ell})$ up to $GL_{2^{n}}(\overline{\mathbb{Q}}_{\ell})$-conjugacy. We have the spin representation relating the two objects \begin{equation} spin: GSpin_{2n + 1} \rightarrow GL_{2^{n}} \end{equation} This defines a map \begin{equation} \Phi(GSpin_{2n + 1}) \rightarrow \Phi(GL_{2^{n}}) \end{equation} The question then is how can one describe the fibers of this map? For example, for $n = 2$, then we have the exceptional isomorphism $GSpin_{5} \simeq GSp_{4}$ and the spin representation is simply the embedding \begin{equation} GSp_{4} \hookrightarrow GL_{4} \end{equation} From this, it easy to see the fibers of the map over a parameter $\phi: \Gamma \rightarrow GL_{4}(\overline{\mathbb{Q}}_{\ell})$ are determined by the similitude character \begin{equation} sim(\phi): \Gamma \xrightarrow{\phi} GSp_{4}(\overline{\mathbb{Q}}_{\ell}) \rightarrow GSp_{4}(\overline{\mathbb{Q}}_{\ell})/Sp_{4}(\overline{\mathbb{Q}}_{\ell}) \simeq \overline{\mathbb{Q}}_{\ell}^{*} \end{equation} where $sim^{2}(\phi) = det(\phi)$. What can one say in general?
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