  In this article, we prove exact estimates for the $W$-invariant Dunkl kernel
and heat kernel, for the root system of type $A$ with arbitrary positive
multiplicities. We apply the estimates of the $W$-invariant Dunkl heat kernel
to compute sharp estimates for the Newton kernel and for the $s$-stable
semigroups generated by a fractional power of the $W$-invariant Dunkl
Laplacian.
