A short proof of the existence of supercuspidal representations for all reductive $p$-adic groups

Raphaël Beuzart-Plessis (1)

1 Institute for Advanced Study

Let $G$ be a reductive $p$-adic group. We give a short proof of the fact that $G$ always admits supercuspidal complex representations. This result has already been established by A. Kret using the Deligne-Lusztig theory of representations of finite groups of Lie type. Our argument is of a different nature and is self-contained. It is based on the Harish-Chandra theory of cusp forms and it ultimately relies on the existence of elliptic maximal tori in $G$.

Mots clés: cusp forms; supercuspidal representations

Domaines: Théorie des représentations [math.RT]; Mathématiques générales [math.GM]

Raphaël Beuzart-Plessis. A short proof of the existence of supercuspidal representations for all reductive $p$-adic groups. Pacific Journal of Mathematics, 2016, 282 (1), pp.27-34. ⟨10.2140/pjm.2016.282.27⟩. ⟨hal-01138463v2⟩