Weyl groups and abelian varieties

Character Varieties of Abelian Groups

We prove that for every reductive group G with a maximal torus T and Weyl group W there is a natural normalization map χ from T/W to an irreducible component, X 0 G(Z ), of the G-character variety XG(Z ) of Z . We prove that this component is reduced and that χ is an isomorphism for classical groups. Additionally, we prove that even though there are no irreducible representations of Z into redu...

Computation of Weyl Groups of G-varieties

Let G be a connected reductive group. To any irreducible G-variety one associates a certain linear group generated by reflections called the Weyl group. Weyl groups play an important role in the study of embeddings of homogeneous spaces. We establish algorithms for computing Weyl groups for homogeneous spaces and affine homogeneous vector bundles. For some special classes of G-varieties (affine...

Griffiths Groups of Supersingular Abelian Varieties

The Griffiths group Grr(X) of a smooth projective variety X over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension r on X modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group Gr(Ak̄) of a supersingular abelian variety Ak̄ over the algebraic closure of a finite field ...

Polarizations of Prym Varieties for Weyl Groups via Abelianization

Let π : Z → X be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group G. For any dominant weight λ consider the curve Y = Z/Stab(λ). The Kanev correspondence defines an abelian subvariety Pλ of the Jacobian of Y . We compute the type of the polarization of the restriction of the canonical principal polarization of Jac(Y ) to P...

Abelian Varieties and Modular Forms on Symplectic Groups