Note on a certain category of mod $p$ representations

1. Problem Statement and Context

Context:
The paper operates within the mod $p$ Langlands program, specifically addressing the classification of smooth admissible mod $p$ representations of $p$-adic groups. The focus is on two cases for a prime $p > 3$ and an integer $f \geq 1$, where $K = \mathbb{Q}_{p^f}$:

1. The $GL_2$-case: Representations of $G_{big} = GL_2(K)$.
2. The Quaternion case: Representations of $G_{big} = D^\times$, where $D$ is the quaternion algebra over $K$ with invariant $1/2$.

The Challenge:
For $f > 1$, the category of smooth admissible mod $p$ representations of $G_{big}$ is "too large" compared to the Galois side, making it difficult to identify which representations correspond to mod $p$ Galois representations.
To address this, a specific full subcategory $\mathcal{C}$ was introduced in recent works ([BHH+23a], [HW24]). This category is defined by specific invariants related to the restriction of a representation $\pi$ to a pro-$p$ subgroup $G_1$ (specifically, the upper-triangular pro-$p$ Iwahori in the $GL_2$ case or a specific unit subgroup in the quaternion case).

The Core Question:
The definition of $\mathcal{C}$ relies on global invariants of the representation. A fundamental open question is whether membership in $\mathcal{C}$ is a local property. Specifically, can one determine if a representation $\pi$ belongs to $\mathcal{C}$ solely by examining its restriction to an arbitrarily small open subgroup $H \subset G_{big}$?

2. Methodology

The author employs a strategy based on completed group rings, $p$-valuations, and associated graded modules to translate the global condition of the category $\mathcal{C}$ into a local condition.

A. Group Structure and $p$-valuations

• The groups $G$ (the quotient of $G_{big}$ by its center) are pro-$p$ groups of dimension $3f$.
• The paper utilizes a saturated $p$-valuation $\omega$ on $G$ that induces the group topology.
• An ordered basis $(g_1, \dots, g_{3f})$ is constructed for $G$. This allows for a homeomorphism $\mathbb{Z}_p^{3f} \to G$ and provides a basis for the completed group ring $\mathbb{F}[[G]]$.

B. Algebraic Framework

• Completed Group Ring: $\mathbb{F}[[G]]$ is analyzed via its associated graded ring $\text{gr } \mathbb{F}[[G]]$ with respect to the maximal ideal $\mathfrak{m}_G$.
• The Category $\mathcal{C}$: A representation $\pi$ is in $\mathcal{C}$ if its dual $\pi^\vee$ (a module over $\mathbb{F}[[G]]$) has an associated graded module $\text{gr } \pi^\vee$ that is annihilated by a power of a specific two-sided ideal $J_C \subset \text{gr } \mathbb{F}[[G]]$.
  • Crucially, the quotient $\text{gr } \mathbb{F}[[G]] / J_C$ is commutative.
• Local Restriction: The author considers an arbitrarily small open subgroup $G_{p^N}$ (the image of $p^N \mathbb{Z}_p^{3f}$ under the ordered basis map). The restriction of $\pi$ to this subgroup corresponds to the structure of $\pi^\vee$ as a module over the subring $\mathbb{F}[[G_{p^N}]]$.

C. The Technical Strategy

The proof proceeds by establishing equivalences between the annihilation condition of the global module and the annihilation condition of modules derived from the restriction to $G_{p^N}$:

1. Graded Subalgebras: The author defines a filtration on $\mathbb{F}[[G_{p^N}]]$ induced by the $p$-valuation, creating a graded subalgebra $\text{gr}_\nu \mathbb{F}[[G_{p^N}]]$.
2. Three Associated Modules: For a module $M$, three graded modules are defined:
   • $\text{gr } M$: The standard associated graded module over $\text{gr } \mathbb{F}[[G]]$.
   • $\text{gr}_{N, int} M$: An intermediate module over $\text{gr}_\nu \mathbb{F}[[G_{p^N}]]$.
   • $\text{gr}_{N, res} M$: A module defined strictly by the restriction to $G_{p^N}$.
3. Ideal Comparison: The author defines a "local" ideal $J_N$ inside $\text{gr}_\nu \mathbb{F}[[G_{p^N}]]$ derived from the global ideal $J_C$.
4. Equivalence Proofs: Through a series of lemmas, the paper proves that $M$ is annihilated by a power of $J_C$ (global condition) if and only if $\text{gr}_{N, res} M$ is annihilated by a power of $J_N$ (local condition).

3. Key Contributions and Results

Main Theorem (Theorem 1):
Let $\pi$ and $\pi'$ be smooth representations of $G_{big}$ with the same central character. If $\pi' \in \mathcal{C}$ and $\pi|_H \cong \pi'|_H$ for some open subgroup $H \subset G_{big}$, then $\pi \in \mathcal{C}$.

Technical Breakdown of the Proof:

• Lemma 1: Establishes that the global annihilation condition ($M \in \mathcal{C}_J$) is equivalent to the global graded module being killed by a power of the local ideal $J_N$.
• Lemma 2: Shows that the condition on the global graded module is equivalent to the condition on the intermediate module $\text{gr}_{N, int} M$.
• Lemma 3 & 4: Prove that the filtration steps allow the transition from the intermediate module to the restricted module $\text{gr}_{N, res} M$. Specifically, it shows that the "distance" between the global filtration and the restricted filtration is bounded (by a factor related to the dimension $4f$), ensuring that annihilation by a power of $J_N$ on the restricted module implies annihilation on the global module.

Generalization:
The result holds for any category defined by a commutative quotient of the associated graded ring, not just the specific $\mathcal{C}$ defined in [BHH+23a] and [HW24].

4. Significance

1. Locality of the Langlands Correspondence: The result confirms that the property of belonging to the "good" category $\mathcal{C}$ (which is conjectured to contain the images of the mod $p$ Langlands correspondence) is a local property. This is a crucial structural insight, suggesting that the "bad" representations (those not in $\mathcal{C}$) can be distinguished by their behavior on arbitrarily small subgroups.
2. Simplification of Verification: It provides a practical tool for researchers. Instead of analyzing global invariants of a representation (which can be computationally intractable), one can verify membership in $\mathcal{C}$ by restricting the representation to a sufficiently small pro-$p$ subgroup.
3. Clarification of the Category $\mathcal{C}$: The paper sheds light on the mysterious nature of $\mathcal{C}$. While it was previously known that $\mathcal{C}$ implies a bound on the Gelfand-Kirillov dimension, this work establishes a robust structural characterization independent of the specific generators of the ideal $J_C$, relying only on the commutativity of the quotient.
4. Applicability: The methodology using $p$-valuations and ordered bases is robust and applies to both the $GL_2$ and Quaternion cases, unifying the treatment of these two distinct settings in the mod $p$ Langlands program.

In summary, Sorgdrager proves that the "good" behavior of mod $p$ representations in the Langlands program is detectable locally, providing a foundational step toward understanding the structure of these representations and potentially constructing the mod $p$ Langlands correspondence for $f > 1$.