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

Abstract

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 $\mathbf{q}$, with denominator depending only on the two-sided cell containing $w$, and dividing 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.