  In this article, we construct the Hodge realization of the polylogarithm
class in the equivariant Deligne-Beilinson cohomology of a certain algebraic
torus associated to a totally real field. We then prove that the de Rham
realization of this polylogarithm gives the Shintani generating class, a
cohomology class generating the values of the Lerch zeta functions of the
totally real field at nonpositive integers. Inspired by this result, we give a
conjecture concerning the specialization of this polylogarithm class at torsion
points, and discuss its relation to the Beilinson conjecture for Hecke
characters of totally real fields.
