Note on a certain category of mod p representations

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\,₂Qpᶠ GL 2 Q p f or of the group of units of the quaternion algebra over Qpᶠ 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 π (with central character) belongs to C C is completely determined by the restriction of π π to an arbitrarily small open subgroup.