On the construction of tame supercuspidal representations

Jessica Fintzen

doi:10.1112/s0010437x21007636

Abstract

Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$. Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$. We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference.

Highlights

In 2001, Yu [Yu01] proposed a construction of smooth complex supercuspidal representations of padic groups that since has been widely used, for example to study the Howe correspondence, to understand distinction of representations of p-adic groups, to obtain character formulas and to construct an explicit local Langlands correspondence
Note that Yu’s construction yields all supercuspidal representations if p does not divide the order of the Weyl group of G [Fin21, Kim07], a condition that guarantees that all tori of G split over a tamely ramified field extension of F
Throughout the paper we fix an additive character φ : F → C∗ of F of conductor P and a reductive group G that is defined over our non-archimedean local field F and that splits over a tamely ramified field extension of F

Summary

Introduction

In 2001, Yu [Yu01] proposed a construction of smooth complex supercuspidal representations of padic groups that since has been widely used, for example to study the Howe correspondence, to understand distinction of representations of p-adic groups, to obtain character formulas and to construct an explicit local Langlands correspondence. Let G be a (connected) reductive group that splits over a tamely ramified field extension of F. Note that Yu’s construction yields all supercuspidal representations if p does not divide the order of the Weyl group of G [Fin, Kim07], a condition that guarantees that all tori of G split over a tamely ramified field extension of F. If G splits over a tamely ramified field extension of F , using (tame) Galois descent, we obtain an embedding of the corresponding Bruhat–Tits buildings B(G , F ) → B(G, F ). Throughout the paper we fix an additive character φ : F → C∗ of F of conductor P and a reductive group G that is defined over our non-archimedean local field F and that splits over a tamely ramified field extension of F. All representations of G(F ) in this paper have complex coefficients and are required to be smooth

Construction of representationsa la Yu

A counterexample