Dual canonical bases and embeddings of symmetric spaces

For a connected reductive group $G_k$ over an algebraically closed field $k$ of char $\mathrm{\ne }2$ and a fixed point subgroup $K_k$ under an algebraic group involution, we construct a quantization and an integral model of any affine embeddings of the symmetric space $G_k\mathrm{/}{K}_k$. We show that the coordinate ring of any affine embedding of $G_k\mathrm{/}{K}_k$ admits a dual canonical basis. We further construct an integral model for the canonical embedding (that is, an embedding which is complete, simple, and toroidal) of $G_k\mathrm{/}{K}_k$. When $G_k$ is of adjoint type, we obtain an integral model for the wonderful compactification of the symmetric space.