Equations Defining Symmetric Varieties and Affine Grassmannians

Suppose $\sigma$ be a simple involution of a semisimple algebraic group $G$ and suppose H is the subgroup of G of points fixed by $\sigma$. If the restricted root system is of type $\mathbf{A,C}$ or $\mathbf{BC}$ and G is simply connected, or if the restricted root system is of type $\mathbf{B}$ and G is of adjoint type, then we describe a standard monomial theory and the equations for the coordinate ring $\mathbb{k}[G/H]$ using the standard monomial theory and the Plücker relations of an appropriate (maybe infinite-dimensional) Grassmann variety.