  In this paper we construct and study the actions of certain deformations of
the Lie algebra of Hamiltonians on the plane on the Chow groups (resp.,
cohomology) of the relative symmetric powers ${\cal C}^{[\bullet]}$ and the
relative Jacobian ${\cal J}$ of a family of curves ${\cal C}/S$. As one of the
applications, we show that in the case of a single curve $C$ this action
induces a integral form of a Lefschetz $\operatorname{sl}_2$-action on the Chow
groups of $C^{[N]}$. Another application gives a new grading on the ring of
0-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin product)
and equips it with an action of the Lie algebra of vector fields on the line.
We also define the groups of tautological classes in $CH^*({\cal
C}^{[\bullet]})$ and in $CH^*({\cal J})$ and prove for them analogs of the
properties established in the case of the Jacobian of a single curve by
Beauville in math.AG/0204188. We also show that the our algebras of operators
preserve the subrings of tautological cycles and act on them via some explicit
differential operators.
