Note on a certain category of mod p representations

Abstract Let p>3 p > 3 be a prime number, f 1 f ≥ 1 an integer. We consider a certain full subcategory C C of the category of smooth admissible mod p representations of either {\, GL\, }₂Q₏㶿 GL 2 Q p f or of the group of units of the quaternion algebra over Q₏㶿 Q p f. This category was introduced in the context of the mod p Langlands program by 1 in the {\, GL\, }₂ GL 2 -case and by 2 in the quaternion case. We prove that whether a smooth admissible mod p representation π (with central character) belongs to C C is completely determined by the restriction of π to an arbitrarily small open subgroup.