  We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB)
connection in genus 1. This is a flat connection over a principal bundle on the
moduli space of elliptic curves with marked points. It restricts to a flat
connection on configuration spaces of points on elliptic curves, which can be
used for proving the formality of the pure braid groups on genus 1 surfaces. We
study the monodromy of this connection and show that it gives rise to a
relation between the KZ associator and a generating series for iterated
integrals of Eisenstein forms. We show that the universal KZB connection
realizes as the usual KZB connection for simple Lie algebras, and that in the
sl_n case this realization factors through the Cherednik algebras. This leads
us to define a functor from the category of equivariant D-modules on sl_n to
that of modules over the Cherednik algebra, and to compute the character of
irreducible equivariant D-modules over sl_n which are supported on the
nilpotent cone.
