We prove a conjecture of Nakajima describing the relation between the geometry of quiver varieties of type A and the geometry of the partial flags varieties and of the nilpotent variety.

Let G be a simple algebraic group and P a parabolic subgroup of G with abelian unipotent radical P u , and let B be a Borel subgroup of G contained in P . Let p u be the Lie algebra of P u and L a Levi factor of P , then L is a Hermitian symmetric subgroup of G and B acts with finitely many orbits both on p u and on G/L. In this paper we study the Bruhat order of the B-orbits in p u and in G/L, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.

Abstract. We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M or over a closed G-stable subvariety of M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spaces and of the closures of the spherical nilpotent orbits. Then we focus on a particular case proving two specific conjectures of Adams, Huang and Vogan on an analogue of the model orbit of the group of type E 8 .

Abstract. 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 σ be the fixed point subgroup. Fix a spherical dominant weight ε in Ω + . We add a node n 0 to the Dynkin diagram of G and, for all simple roots α, we join n 0 with the node n α of the simple root α by ε(α ∨ ) lines, and we put an arrow in direction of n α if ε(α ∨ ) ≥ 2. In the cases relevant for us, the Kac-Moody group e 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 varietyP . The homogeneous coordinate ring Γ Gr = j≥0 Γ(Gr, L j ) 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

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