  In this paper, we look at the problem of determining the composition factors
for the affine graded Hecke algebra via the computation of Kazhdan-Lusztig type
polynomials. We review the algorithms of \cite{L1,L2}, and use them in
particular to compute, at every real central character which admits tempered
modules, the geometric parameterization, the Kazhdan-Lusztig polynomials, the
composition series, and the Iwahori-Matsumoto involution for the
representations with Iwahori fixed vectors of the split $p$-adic groups of type
$G_2$ and $F_4$.
