On the ring of cooperations for real hermitian K-theory

Let kq denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations ${\pi }_{\ast \ast }^{\mathbb{R}}(\text{kq}\otimes \text{kq})$ in the stable motivic homotopy category $\text{SH}(\mathbb{R})$, giving a full description in terms of Brown–Gitler comodules. To do this, we decompose the $E_2$-page of the motivic Adams spectral sequence and show that it must collapse. The description of the $E_2$-page is accomplished by a series of algebraic Atiyah–Hirzebruch spectral sequences which converge to the summands of the $E_2$-page. Along the way, we prove a splitting result for the very effective symplectic K-theory ksp over any base field of characteristic not two.