Coordinate rings on symmetric spaces

Abstract Let $$G_k$$ G k be a connected reductive group over an algebraically closed field k of char $$\ne 2$$ ≠ 2 . Let $$\theta _k$$ θ k be an algebraic group involution of $$G_k$$ G k and denote the fixed point subgroup by $$K_k$$ K k . We construct an integral model for the symmetric space $$K_k \backslash G_k$$ K k \ G k with a natural action of the Chevalley group scheme over integers. We show the coordinate ring $$k[K_k \backslash G_k]$$ k [ K k \ G k ] admits a canonical basis, as well as a good filtration as a $$G_k$$ G k -module. We also construct a canonical basis and an integral form for the space of $$K_k$$ K k -biinvariant functions on $$k[G_k]$$ k [ G k ] . Our results rely on the construction of quantized coordinate algebras of symmetric spaces, using the theory of canonical bases on quantum symmetric pairs.