The $RO(C_2)$-graded Hurewicz image of Atiyah's Real K-theory

We determine the $RO(C_2)$-graded $k\mathbb{R}$-Hurewicz image, where $k\mathbb{R}$ denotes the connective Atiyah’s Real $K$-theory spectrum. As a consequence, we obtain four infinite families in the $RO(C_2)$-graded equivariant stable stems that are $C_2$-equivariant refinements of the classical $\mu$-families. Using a $C_2$-equivariant $v_{1,0}$-self map, we explicitly construct an equivariant Adams periodicity operator $P_{C_2}$ that captures these $v_1$-periodicity phenomena in the $C_2$-stable stems.