Note on a certain category of mod p representations

Abstract Let $$p&gt;3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> be a prime number, $$f\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> an integer. We consider a certain full subcategory $$\mathcal {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> of the category of smooth admissible mod p representations of either $${{\,\textrm{GL}\,}}_2{\textbf{Q}}_{p^f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>GL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>Q</mml:mi> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>f</mml:mi> </mml:msup> </mml:msub> </mml:mrow> </mml:math> or of the group of units of the quaternion algebra over $${\textbf{Q}}_{p^f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>f</mml:mi> </mml:msup> </mml:msub> </mml:math> . This category was introduced in the context of the mod p Langlands program by [1] in the $${{\,\textrm{GL}\,}}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mspace/> <mml:mtext>GL</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -case and by [2] in the quaternion case. We prove that whether a smooth admissible mod p representation $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> (with central character) belongs to $$\mathcal {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> is completely determined by the restriction of $$\pi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> to an arbitrarily small open subgroup.