We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive $p$-adic group tend to zero uniformly for every noncentral semisimple element.

中文翻译：

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Supercuspidal 表示的渐近行为和家庭的 Sato-Tate 均衡分布

当我们在给定的有限素数集上改变规定的超尖点表示时，我们建立了自守表示族的属性。对于由 J.-K. 于我们用误差项证明极限重数性质。因此，我们获得了这些家族的 Hecke 特征值的 Sato-Tate 等分布。主要的新成分是表明，对于每个非中心半单元，在连接的还原 $p$-adic 群上，随着形式度的增加，驯服的超尖点表示的矩阵系数的轨道积分趋于一致为零。