Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs

Let $G_k$ be a connected reductive algebraic group over an algebraically closed field $k$ of characteristic $\neq 2$. Let $K_k \subset G_k$ be a quasi-split symmetric subgroup of $G_k$ with respect to an involution $\theta_k$ of $G_k$. The classification of such involutions is independent of the characteristic of $k$ (provided not $2$). We first construct a closed subgroup scheme $\mathbf{G}^\imath$ of the Chevalley group scheme $\mathbf{G}$ over $\mathbb{Z}$. The pair $(\mathbf{G}, \mathbf{G}^\imath)$ parameterizes symmetric pairs of the given type over any algebraically closed field of characteristic $\neq 2$, that is, the geometric fibre of $\mathbf{G}^\imath$ becomes the reductive group $K_k \subset G_k$ over any algebraically closed field $k$ of characteristic $\neq 2$. As a consequence, we show the coordinate ring of the group $K_k$ is spanned by the dual $\imath$canonical basis of the corresponding $\imath$quantum group. We then construct a quantum Frobenius splitting for the quasi-split $\imath$quantum group at roots of $1$. This generalizes Lusztig's quantum Frobenius splitting for quantum groups at roots of $1$. Over a field of positive characteristic, our quantum Frobenius splitting induces a Frobenius splitting of the algebraic group $K_k$. Finally, we construct Frobenius splittings of the flag variety $G_k / B_k$ that compatibly split certain $K_k$-orbit closures over positive characteristics. We deduce cohomological vanishings of line bundles as well as normalities. Results apply to characteristic $0$ as well, thanks to the existence of the scheme $\mathbf{G}^\imath$. Our construction of splittings is based on the quantum Frobenius splitting of the corresponding $\imath$quantum group.