  Let $\tilde{W}$ be an extended affine Weyl group, $\mathbf{H}$ be the
corresponding affine Hecke algebra over the ring
$\mathbb{C}[\mathbf{q}^\frac{1}{2}, \mathbf{q}^{-\frac{1}{2}}]$, and $J$ be
Lusztig's asymptotic Hecke algebra, viewed as a based ring with basis
$\{t_w\}$. Viewing $J$ as a subalgebra of the
$(\mathbf{q}^{-\frac{1}{2}})$-adic completion of $\mathbf{H}$ via Lusztig's map
$\phi$, we use Harish-Chandra's Plancherel formula for $p$-adic groups to show
that the coefficient of $T_x$ in $t_w$ is a rational function of $\mathbf{q}$,
depending only on the two-sided cell containing $w$, and with no poles outside
of a finite set of roots of unity depending only on $\tilde{W}$. For $w$ in the
lowest two-sided cell, and for all cells in types $\tilde{A}_n$, $\tilde{C}_2$,
and $\tilde{G}_2$, we show that the denominators all divide a power of the
Poincar\'{e} polynomial of the finite Weyl group. As an application, we
conjecture that these denominators encode more detailed information about the
failure of the Kazhdan-Lusztig classification at roots of the Poincar\'{e}
polynomial than is currently known.
  Along the way, we show that upon specializing $\mathbf{q}=q>1$, the map from
$J$ to the Harish-Chandra Schwartz algebra is injective. As an application of
injectivity, we give a novel criterion for an Iwahori-spherical representation
to have fixed vectors under a larger parahoric subgroup in terms of its
Kazhdan-Lusztig parameter.
