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

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.