View the orthogonal and special orthogonal groups $O(n)$ and $SO(n)$ either as algebraic groups over $\mathbb{C}$ or as compact Lie groups. The representations and representation rings of both are known. Consider however a direct product $\prod_iO(n_i)$ and the closed subgroup $G$ of $\prod_iO(n_i)$ consisting of elements $(g_i)_i$ such that $\prod_i\mathrm{det}g_i=1$. What is the relationship between representations of $G$, representation of $\prod_iO(n_i)$, and representations of $\prod_iSO(n_i)$?
This paper of Segal implies that the representation ring of $G$ will be a direct product, with half as many factors as $R(\prod_iO(n_i))$, but is there really no simple explicit answer?