The real Brown-Peterson homology of ${\mathrm{\Omega }}^{\rho }{S}^{\rho +1}$

We compute the $RO(C_2)$-graded real Brown–Peterson homology of the representation-loop space ${\mathrm{\Omega }}^{\rho }{S}^{\rho +1}$, where $\rho$ is the regular representation of the cyclic group of order two. This calculation gives a $C_2$-equivariant analogue of the classical computation of Brown–Peterson homology of the double loop space ${\mathrm{\Omega }}^2{S}^3$ due to Ravenel. Along the way, we develop comodule Nishida relations for $\rho$-loop spaces.