The asymptotic Hecke algebra and rigidity

We reprove the surjectivity statement of Braverman-Kazhdan's spectral description of Lusztig's asymptotic Hecke algebra $J$ in the context of $p$-adic groups. The proof is based on Bezrukavnikov-Ostrik's description of $J$ in terms of equivariant $K$-theory. As a porism, we prove that the action of $J$ extends from the non-strictly positive unramified characters to the complement of a finite union of divisors, and that the trace pairing between the Ciubotaru-He rigid cocentre of an affine Hecke algebra with equal parameters and the rigid quotient of its Grothendieck group is perfect whenever the parameter $q$ is not a root of the Poincar\'{e} polynomial of the finite Weyl group. Without recourse to $K$-theory, we prove a weak version of Xi's description of $J$ in type $A$. As an application of relationship between $J$ and the rigid cocentre, we prove that the formal degree of a unipotent discrete series representation of a connected reductive $p$-adic group $G$ with a split inner form has denominator dividing the Poincar\'{e} polynomial of the Weyl group of $G$. Additionally, we give formulas for $t_w$ in terms of inverse and spherical Kazhdan-Lusztig polynomials for $w$ in the lowest cell.