Given an algebraic cycle on a smooth projective variety X over C, its fundamental class belongs to the group of Hodge classes in cohomology of X; the Hodge Conjecture says that, conversely, all Hodge classes are algebraic. In the mid-1980s, Kumar Murty proved that the Hodge conjecture is true when X is any power of an abelian variety which is nondegenerate with no simple factor of Albert type III. He did this by exploiting the intimate relationship between algebraic cycles and the representation theory of certain reductive groups. In this talk we describe another consequence of this connection.

An algebraic cycle with *trivial* fundamental class leads to an extension of Hodge-theoretic data, and in a variational context to a family of extensions called a normal function. These may be viewed as `horizontal' sections of a bundle of complex tori, and are used to detect cycles modulo algebraic (or rational) equivalence. Conversely, the existence of normal functions can be used to predict that interesting cycles are present. . . or absent: a famous theorem of Green and Voisin states that for projective hypersurfaces of large enough degree, there are no normal functions (into the intermediate Jacobian bundle associated to these hypersurfaces) over any etale neighborhood of the coarse moduli space.

Inspired by recent work of Friedman-Laza on Hermitian variations of Hodge structure and Oort's conjecture on special (i.e. Shimura) subvarieties in the Torelli locus, R. Keast and I wondered about the existence of normal functions over etale neighborhoods of Shimura varieties. Here the function is supposed to take values in a family of intermediate Jacobians associated to a representation of a reductive group, and in many cases to families of abelian varieties. In this talk I will explain our classification of the cases where a Green-Voisin analogue does *not* hold and where one therefore expects interesting cycles to occur, and give some evidence that these predictions might be `sharp'.