Note on a certain category of mod p representations

Abstract Let $$p>3$$ p > 3 be a prime number, $$f\ge 1$$ f ≥ 1 an integer. We consider a certain full subcategory $$\mathcal {C}$$ C of the category of smooth admissible mod p representations of either $${{\,\textrm{GL}\,}}_2{\textbf{Q}}_{p^f}$$ GL 2 Q p f or of the group of units of the quaternion algebra over $${\textbf{Q}}_{p^f}$$ Q p f . This category was introduced in the context of the mod p Langlands program by [1] in the $${{\,\textrm{GL}\,}}_2$$ GL 2 -case and by [2] in the quaternion case. We prove that whether a smooth admissible mod p representation $$\pi $$ π (with central character) belongs to $$\mathcal {C}$$ C is completely determined by the restriction of $$\pi $$ π to an arbitrarily small open subgroup.