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Let $\tilde{W}$ be an extended affine Weyl group, $\mathbf{H}$ be thecorresponding affine Hecke algebra over the ring$\mathbb{C}[\mathbf{q}^\frac{1}{2}, \mathbf{q}^{-\frac{1}{2}}]$, and $J$ beLusztig'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 showthat 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 outsideof a finite set of roots of unity depending only on $\tilde{W}$. For $w$ in thelowest 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 thePoincar\'{e} polynomial of the finite Weyl group. As an application, weconjecture that these denominators encode more detailed information about thefailure 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 ofinjectivity, we give a novel criterion for an Iwahori-spherical representationto have fixed vectors under a larger parahoric subgroup in terms of itsKazhdan-Lusztig parameter.

Denominators in Lusztig's asymptotic Hecke algebra via the Plancherel formula