Coordinate rings on symmetric spaces

Let Gₖ be a connected reductive group over an algebraically closed field k of char 2. Let ₖ be an algebraic group involution of Gₖ and denote the fixed point subgroup by Kₖ. We construct an integral model for the symmetric space Kₖ Gₖ with a natural action of the Chevalley group scheme over integers. We show the coordinate ring kKₖ Gₖ admits a canonical basis, as well as a good filtration as a Gₖ-module. We also construct a canonical basis and an integral form for the space of Kₖ-biinvariant functions on kGₖ. Our results rely on the construction of quantized coordinate algebras of symmetric spaces, using the theory of canonical bases on quantum symmetric pairs.