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