Projective representation (awaiting review)

To each mod-$\ell$ Galois representation $\rho:\Gal_K \to G(\F_{\ell})$, with values in the $\F_{\ell}$-points of an algebraic group $G$, is associated a projective representation $\mathbb{P}\rho:\Gal_K \to G(\F_{\ell})/\F_{\ell}^*$, defined by composing $\rho$ with the projection $G(\F_{\ell}) \to G(\F_{\ell})/\F_{\ell}^*$.

In case $\ell=2$, the projective representation is the same as the original representation.