The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from its Satake diagram, in a way that is suited for the use with computer algebra systems; an example implementation for Maple Version 10 can be fou...

A Hodge structure V of weight k on which a CM eld acts de nes, under certain conditions, a Hodge structure of weight k 1, its half twist. In this paper we consider hypersurfaces in projective space with a cyclic automorphism which de nes an action of a cyclotomic eld on a Hodge substructure in the cohomology. We determine when the half twist exists and relate it to the geometry and moduli of th...

Let h2 = Sp(4,R)/U(2) be the Siegel upperhalf space of rank 2. The quotient Sp(4,Z)\h2 has three remarkable properties: (a) it is the moduli space of principally polarized abelian surfaces, (b) it has the structure of a quasi-projective complex algebraic variety which is defined over the rational numbers Q, and (c) it has a natural compactification (the BailyBorel Satake compactification) which...

The endomorphism algebra of a K3 type Hodge structure is a totally real field or a CM field. In this paper we give a low brow introduction to the case of a totally real field. We give existence results for the Hodge structures, for their polarizations and for certain K3 surfaces. We consider the Kuga Satake variety of these Hodge structures and we discuss some examples. Finally we indicate vari...

Abstract. For a split reductive group scheme Ǧ over a commutative ring k with Weyl group W , there is an important functor Rep(Ǧ, k) → Rep(W, k) defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dua...

We state a conjecture about the Weyl group action coming from Geometric Satake on zero-weight spaces in terms of equivariant multiplicities Mirković-Vilonen cycles. prove it for small coweights type A. In this case, using work Braverman, Gaitsgory and Vybornov, we show that basis agrees with Springer basis. rephrase Joseph Hotta. also have analogous results Ginzburg's Lagrangian construction sl...

Let h2 = Sp(4,R)/U(2) be the Siegel upperhalf space of rank 2. The quotient Sp(4,Z)\h2 has three remarkable properties: (a) it is the moduli space of principally polarized abelian surfaces, (b) it has the structure of a quasi-projective complex algebraic variety which is defined over the rational numbers Q, and (c) it has a natural compactification (the BailyBorel Satake compactification) which...