  We study the problem of L^p-boundedness (1 < p < \infty) of operators of the
form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant
differential operators L_1,...,L_n on a Lie group G of polynomial growth, which
generate an algebra containing a weighted subcoercive operator. In particular,
when G is a homogeneous group and L_1,...,L_n are homogeneous, we prove
analogues of the Mihlin-H\"ormander and Marcinkiewicz multiplier theorems.
