Equations Defining Symmetric Varieties and Affine Grassmannians

Let σ be a simple involution of an algebraic semisimple group G and let H be the subgroup of G of points fixed by σ. 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 Plücker relations of an appropriate (maybe infinite dimensional) Grassmann variety. The aim of this paper is the description of the coordinate ring of the symmetric varieties and of certain rings related to their wonderful compactification. The main tool to achieve this goal is a (possibly infinite dimensional) Grassmann variety associated to a pair consisting of a symmetric space and a spherical representation. More precisely, let G be a semisimple algebraic group over an algebraically closed field k of characteristic 0 and let σ be a simple involution of G (i.e. G⋊{id, σ} acts irreducibly on the Lie algebra of G). Let H = G be the fixed point subgroup. The quotient G/H is an affine variety, called a symmetric variety. A simple finite dimensional G-module V is called spherical (for H) if V H 6= 0. By results of Helgason [9] and Vust [24], these modules are parametrized by a submonoid Ω of the dominant weights of a suitable root system, called the restricted root system. As a G-module, k[G/H ] is well understood: it is the direct sum ⊕ V spherical V . Fix a spherical dominant weight ε in Ω. We add a node n0 to the Dynkin diagram of G and, for all simple roots α, we join n0 with the node nα of the simple root α by ε(α ) lines, and we put an arrow in direction of nα if ε(α ) ≥ 2. In the cases relevant for us, the Kac-Moody group G associated to the extended diagram will be of finite or affine type. Let L be the ample generator of Pic(Gr) for the generalized Grassmann variety Gr = G/P . The homogeneous coordinate ring ΓGr = ⊕ j≥0 Γ(Gr,L ) is the quotient of the symmetric algebra S(Γ(Gr,L)) by an ideal generated by quadratic relations, the generalized Plücker relations. Since our aim is to relate these Plücker relations to k[G/H ], we say that the monoid Ω is quadratic if (it is free and) its basis has the following property with respect to the dominant order of the restricted root system: any element of Ω that is less than the 2000 Mathematics Subject Classification. 14M15, 14M17, 17B10, 13F50.