  We prove Lusztig's conjectures ${\bf P1}$--${\bf P15}$ for the affine Weyl
group of type $\tilde{G}_2$ for all choices of parameters. Our approach to
compute Lusztig's $\mathbf{a}$-function is based on the notion of a "balanced
system of cell representations" for the Hecke algebra. We show that for
arbitrary Coxeter type the existence of balanced system of cell representations
is sufficient to compute the $\mathbf{a}$-function and we explicitly construct
such a system in type $\tilde{G}_2$ for arbitrary parameters. We then
investigate the connection between Kazhdan-Lusztig cells and the Plancherel
Theorem in type $\tilde{G}_2$, allowing us to prove ${\bf P1}$ and determine
the set of Duflo involutions. From there, the proof of the remaining
conjectures follows very naturally, essentially from the combinatorics of Weyl
characters of types $G_2$ and $A_1$, along with some explicit computations for
the finite cells.
