$MSp$ localized away from $2$ and odd formal group laws

We investigate the relationship between complex and symplectic cobordism localized away from the prime~$2$ and show that these theories are related much as a real Lie group is related to its complexification. This suggests that ideas from the theory of symmetric spaces might be used to illuminate these subjects. In particular, we give an explicit equivalence of ring spectra $$MSp[1/2]\wedge Sp/U_+\simeq MU[1/2]$$ and deduce that $MU[1/2]$ is a wedge of copies of $MSp[1/2]$. We discuss the implications for the structure of the stable operation algebra $MSp[1/2]^*MSp[1/2]$ and the dual cooperation algebra $MSp[1/2]_*MSp[1/2]$. Finally we describe some related Witt vector algebra and apply our results to the study of formal involutions on the category of formal group laws over a $\mathbb{Z}[1/2]$-algebra.