Let G be a semisimple algebraic
group of hermitian type defined over Q, let X ≃ GR∕K, where K ⊂ G is a maximal
compact subgroup, be the symmetric domain associated to G, let Γ be an
arithmetic subgroup of G, let (π,E) be a finite-dimensional representation of
G defined over Q, let 𝒰 := Γ ∖ X, and let ℰ be the locally constant sheaf
over 𝒰 associated to (π,E). Then under certain conditions on G, Γ and
(π,E), the quotient 𝒰 is a complex projective variety and there exists a Kuga
fiber variety 𝒱, i.e., a complex projective variety with the structure of an
analytic family of abelian varieties parametrized by 𝒰, such that Ha(𝒰;ℰ) may
be identified with a subspace of H∗(𝒱;Q). The purpose of this paper is to
show that for a certain class of nontrivial (π,E) the subspace of H∗(𝒱;Q)
with which Ha(𝒰;ℰ) is identified is algebraically defined, or in other words
that this subspace is contained in Hr(𝒱;Q) for some r and a projection
from Hr(𝒱;Q) to it is induced by an algebraic class in H∗(𝒱×𝒱;Q). In
particular, since the projection of an algebraic class in Hr(𝒱;Q) is again an
algebraic class, this paper provides an answer to the question of how to define
algebraic classes in Ha(𝒰;ℰ) for some nontrivial local coefficient systems
ℰ.
