Dual canonical bases and embeddings of symmetric spaces

For a connected reductive group Gₖ over an algebraically closed field k of char ≠ 2 and a fixed point subgroup Kₖ under an algebraic group involution, we construct a quantization and an integral model of any affine embeddings of the symmetric space Gₖ/Kₖ. We show that the coordinate ring of any affine embedding of Gₖ/Kₖ 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 Gₖ/Kₖ. When Gₖ is of adjoint type, we obtain an integral model for the wonderful compactification of the symmetric space.