Note on a certain category of mod $p$ representations

Let $p>3$ be a prime number, $f\geq1$ an integer. We consider a certain full subcategory $\mathcal C$ of the category of smooth admissible mod $p$ representations of either $\text{GL}2\mathbf Q{p^f}$ or of the group of units of the quaternion algebra over $\mathbf Q_{p^f}$. This category was introduced in the context of the mod $p$ Langlands program by Breuil-Herzig-Hu-Morra-Schraen in the $\text{GL}_2$-case and by Hu-Wang in the quaternion case. We prove that whether a smooth admissible mod $p$ representation $π$ (with central character) belongs to $\mathcal C$ is completely determined by the restriction of $π$ to an arbitrarily small open subgroup.

💡 Research Summary

The paper studies smooth admissible mod p representations of two p‑adic Lie groups that arise from a local field K = ℚ_{p^f} with p > 3 and f ≥ 1. The first group is the quotient I₁/Z₁ where I₁ is the upper‑triangular pro‑p Iwahori subgroup of GL₂(K) and Z₁ its scalar centre; the second is the quotient U₁/Z₁ where U₁ = 1 + Π 𝒪_D is the principal congruence subgroup of the unit group of the quaternion algebra D over K (Π is a uniformiser with Π² = p) and Z₁ = 1 + p 𝒪_K. Both groups are pro‑p of dimension 3f and have trivial action of the centre on any representation with a fixed central character.

In earlier work (Breuil‑Herzig‑Hu‑Morra‑Schraen for GL₂ and Hu‑Wang for the quaternion case) a full subcategory 𝒞 of the category of smooth admissible mod p representations of the larger groups GL₂(K) or D^× was introduced. An object π belongs to 𝒞 precisely when the graded module gr π^∨, obtained from the m_G‑adic filtration on the Pontryagin dual π^∨ (viewed as a pseudocompact module over the completed group ring F⟦G⟧), is annihilated by a power of a distinguished two‑sided ideal J_𝒞 ⊂ gr F⟦G⟧. In the GL₂‑case J_𝒞 is the ideal I_G defined in