On some 𝑝-adic and mod 𝑝 representations of quaternion algebra over ℚ𝑝

Let 𝐷 be the nonsplit quaternion algebra over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="double-struck">Q</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> Qₚ . We prove that a class of admissible unitary Banach space representations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>D</m:mi> <m:mo>×</m:mo> </m:msup> </m:math> D× of global origin are topologically of finite length.