  Let W be a Coxeter group and L be a weight function on W. Following Lusztig,
we have a corresponding decomposition of W into left cells, which have
important applications in representation theory. We study the case where $W$ is
an affine Weyl group of type $\tilde{G_{2}}$. Using explicit computation with
\textsf{CHEVIE}, we show that (1) there are only finitely many possible
decompositions into left cells and (2) the number of left cells is finite in
each case, thus confirming some of Lusztig's conjectures in this case. For the
proof, we show some equalities on the Kazhdan-Lusztig polynomials which hold
for any affine Weyl groups.
