  We construct a flat connection on the elliptic configuration space associated
to any complex semisimple Lie algebra g. This elliptic Casimir connection has
logarithmic singularities, and takes values in the deformed double current
algebra of g defined by Guay. It degenerates to the trigonometric Casimir
connection of g constructed by the first author. By analogy with the rational
and trigonometric cases, we conjecture that the monodromy of the elliptic
Casimir connection is described by the quantum Weyl group operators of the
quantum toroidal algebra of g.
