DENOMINATORS IN LUSZTIG’S ASYMPTOTIC HECKE ALGEBRA VIA THE PLANCHEREL FORMULA

Abstract Let $W_{\mathrm {aff}}$ 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 Poincaré 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 Poincaré 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.