We prove Lusztig's conjectures P1–P15 for the affine Weyl group of type˜G2type˜ type˜G2 for all choices of parameters. Our approach is based on the notion of a " balanced system of representations " for the Hecke algebra. Moreover we show that for arbitrary Coxeter type the existence of such a system is sufficient to compute Lusztig's a-function. We also describe connections between the cell decomposition and the structure of the Plancherel Theorem in type˜G2type˜ type˜G2.