Equations defining symmetric varieties and affine Grassmannians

Abstract

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