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Dylan Wilson has constructed a $C_p$-equivariant space, $\mathbb{CP}^{\infty}_{\mu_p}$, which generalizes the complex conjugation action on projective space. I will explain several different ways of viewing this space, as well as the associated notion of a $\mu_p$-orientation. In particular, I will discuss $\mu_p$-orientations of height $p-1$ Morava $E$-theories, as well as a $\mu_3$-orientation of tmf(2). I will describe how a single element $v_1^{mu_p}$ in the stable homotopy of $CP^{\infty}_{\mu_p}$ determines the $C_p$-action on the homotopy groups of height $p-1$ Morava $E$-theory. This is joint work with Andrew Senger and Dylan Wilson.