Exotic branes and mixed-symmetry potentials II: Duality rules and exceptional p-form gauge fields

In $U$-duality-manifest formulations, supergravity fields are packaged into covariant objects such as the generalized metric and $p$-form fields $\mathcal A_p^{I_p}$. While a parameterization of the generalized metric in terms of supergravity fields is known for $U$-duality groups $E_n$ with $n\leq 8$, a parameterization of $\mathcal A_p^{I_p}$ has not been fully determined. In this paper, we propose a systematic method to determine the parameterization of $\mathcal A_p^{I_p}$, which necessarily involves mixed-symmetry potentials. We also show how to systematically obtain the $T$- and $S$-duality transformation rules of the mixed-symmetry potentials entering the multiplet. As the simplest non-trivial application, we find the parameterization and the duality rules associated with the dual graviton. Additionally, we show that the 1-form field $\mathcal A_1^{I_1}$ can be regarded as the generalized graviphoton in the exceptional spacetime.