We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.

中文翻译：

---

常规超尖牙表示

我们表明，在良好的残差特征中，驯服的分枝还原 p-进群 G 的大多数超尖峰表示来自对 (S,\theta)，其中 S 是 G 的驯服椭圆最大环面，\theta 是S 满足简单的根论性质。然后，我们为出现在这些超尖峰表示的 Adler-DeBacker-Spice 字符公式中的统一根给出了一个新表达式，并使用它来表明该公式与实还原群的离散级数表示的字符公式有惊人的相似之处. 在此引导下，我们明确地为这些超尖牙表示构建了局部朗兰兹对应，并证明了在 tooral 表示的情况下的稳定性和内窥镜转移。