[Paper Review] Diagrammes canoniques et representations modulo $p$ de $\GL_2(F)$

Yongquan Hu|arXiv (Cornell University)|Apr 26, 2009

Advanced Algebra and Geometry8 references3 citations

TL;DR

This paper introduces a canonical diagram construction for irreducible smooth representations of $\mathrm{GL}_2(F)$ over $\overline{\mathbb{F}}_p$, where $F$ is a non-Archimedean local field of residual characteristic $p$. It proves that the original representation is completely determined up to isomorphism by this diagram, and explicitly computes the diagram in key cases, particularly when $F = \mathbb{Q}_p$ or $\pi$ is non-supersingular.

ABSTRACT

Let $p$ be a prime number and $F$ a local field with residual characteristic $p$. In this article, to an irreducible smooth representation of $GL_2(F)$ over $\bar{\mathbf{F}}_p$ with central character, we associate canonically a diagram which determines the original representation up to isomorphism. We also determine it in some cases.

Motivation & Objective

• To establish a canonical diagram invariant that classifies irreducible smooth $\mathrm{GL}_2(F)$-representations over $\overline{\mathbb{F}}_p$ with central character.
• To extend the diagrammatic framework of Paškūnas to explicitly reconstruct representations from their diagram invariants.
• To clarify the link between finiteness properties of the diagram and admissibility or finite presentation of the representation.
• To compute the canonical diagram explicitly in non-supersingular and $F = \mathbb{Q}_p$ cases.
• To investigate the structure of $I_1$-invariants and their role in determining the diagram components.

Proposed method

• Define the canonical diagram $D(\pi) = (D_0(\pi), D_1(\pi), \mathrm{can})$ where $D_1(\pi) = \pi^{I_1}$ and $D_0(\pi) = \langle KZ \cdot D_1(\pi) \rangle$.
• Use the functor $H_0$ from the category of diagrams to smooth $G$-representations, showing $H_0(D(\pi)) \cong \pi$.
• Apply the theory of coefficient systems on the Bruhat-Tits tree and $I_1$-invariants to analyze the structure of $D_1(\pi)$.
• Utilize the action of Hecke operators $T$ and $S$ on representations to compute invariants and filtrations.
• Leverage the reciprocity of Frobenius and the structure of compact induction to relate $D_1(\pi)$ to $\pi$-invariants.
• Use explicit computations in the case $F = \mathbb{Q}_p$ and for non-supersingular representations to verify the diagram reconstruction.

Experimental results

Research questions

• RQ1 Can an irreducible smooth $\mathrm{GL}_2(F)$-representation over $\overline{\mathbb{F}}_p$ with central character be reconstructed up to isomorphism from a canonical diagram?
• RQ2 What is the precise structure of $D_1(\pi)$, the $I_1$-invariant subspace, in terms of the original representation $\pi$?
• RQ3 How do finiteness conditions on the diagram components relate to admissibility and finite presentation of $\pi$?
• RQ4 Under what conditions does $D_1(\pi) = \pi^{I_1}$ hold, and when does it equal $\pi$?
• RQ5 Can the canonical diagram be explicitly computed in the case $F = \mathbb{Q}_p$ or for non-supersingular representations?

Key findings

• There is a natural isomorphism $H_0(D(\pi)) \cong \pi$, proving that the canonical diagram $D(\pi)$ determines $\pi$ up to isomorphism.
• $D_1(\pi) = \pi^{I_1}$ always holds, and $D_1(\pi) = \pi$ if $F$ has characteristic $p$ and $\pi$ is supersingular.
• For $F = \mathbb{Q}_p$, $D_1(\pi) = \pi^{I_1}$ holds, and the diagram is explicitly computable.
• The conditions that $D_1(\pi)$ is finite-dimensional, $\pi$ admits a finite presentation, and $\pi$ admits a standard presentation are all equivalent.
• If any of these finiteness conditions hold, then $\pi$ is admissible and the space $I^+(\pi)^{\left(\begin{smallmatrix}1&\mathcal{O}\\0&1\end{smallmatrix}\right)}$ is finite-dimensional.
• In the non-supersingular case, $D_1(\pi) = \pi^{I_1}$, and the diagram is constructed from $I_1$-invariants and their $KZ$-span.