Lifting banal representations of classical groups

1. Problem Statement and Context

The paper addresses a fundamental gap in the representation theory of reductive groups over local non-archimedean fields ($F$). Specifically, it investigates the relationship between representations with coefficients in a field of characteristic zero ($\mathbb{Q}_\ell$) and those in a field of positive characteristic ($\mathbb{F}_\ell$), where $\ell$ is a prime not dividing the residue characteristic of $F$.

• The Core Question: Given a smooth irreducible representation $\pi$ of a classical group $G(F)$ over $\mathbb{F}_\ell$, does there exist a "lift" $\tilde{\pi}$ over $\mathbb{Q}_\ell$ such that the reduction of an integral structure of $\tilde{\pi}$ modulo $\ell$ is isomorphic to $\pi$?
• The "Banal" Condition: The paper focuses on the case where $\ell$ is a banal prime. A prime $\ell$ is banal with respect to $G$ if it does not divide the cardinality of the group of points over the residue field, $|G(k)|$. In this regime, the representation theory over $\mathbb{F}_\ell$ is expected to behave similarly to the complex case ($\mathbb{C}$), unlike the "non-banal" (or "bad") case where modular representation theory becomes significantly more complex.
• Previous Work: While this lifting property was established for general linear groups ($GL_n$) and their inner forms (e.g., by Mínguez and Sécherre), the behavior for classical groups (symplectic, orthogonal, unitary) remained largely unresolved.

2. Methodology

The author employs a recursive construction strategy, building representations from cuspidal data and verifying that the lifting property is preserved at each step. The methodology relies on four main pillars:

A. Intertwining Operators

The paper utilizes the theory of intertwining operators developed over arbitrary algebraically closed fields (following [7]).

• Mechanism: For a parabolic subgroup $P$ with Levi factor $M$, and an admissible representation $\pi$ of $M$, intertwining operators $J_{P,P'}(\pi)$ map induced representations to one another.
• Key Lemma (Lemma 3.7): The author proves that under specific regularity conditions (vanishing of certain poles/zeros of the operators), the image of an integral lattice under an intertwining operator remains integral, and its reduction modulo $\ell$ corresponds to the image of the reduced operator. This allows the transfer of irreducibility and lifting properties from $\mathbb{Q}_\ell$ to $\mathbb{F}_\ell$.

B. Derivatives

The paper adapts the theory of derivatives (originally for $GL_n$ by Bernstein-Zelevinsky and extended to classical groups) to the modular setting.

• Concept: A representation $\pi$ is $\rho$-reduced if it does not contain specific subquotients related to a cuspidal representation $\rho$. The derivative $D_\rho(\pi)$ extracts the "core" representation after removing $\rho$-factors.
• Application: This allows the author to reduce the problem of lifting an arbitrary representation to the problem of lifting a "reduced" representation, effectively stripping away complexity recursively.

C. Arthur Classification and Tempered Representations

For classical groups, the author leverages the Arthur classification of tempered representations.

• Reduction: The problem is reduced to understanding representations induced from self-dual cuspidal representations of Levi subgroups.
• Progenerators: Using the work of [19] and [28], the author analyzes the endomorphism algebras of induced representations. In the banal case, cuspidal representations are projective in their respective blocks. This allows the use of progenerators to describe the category of representations, ensuring that the structure of the representation over $\mathbb{F}_\ell$ mirrors that over $\mathbb{Q}_\ell$.

D. MVW-Involutions

The paper utilizes the MVW-involution (a covariant involution on the category of representations) to relate a representation to its dual. This is crucial for handling the self-duality conditions inherent in classical groups (symplectic/orthogonal/unitary) which differ from the general linear case.

3. Key Contributions and Results

Main Theorem (Theorem 1 / Corollary 6.3)

Statement: Let $G$ be a symplectic or split orthogonal group over a local non-archimedean field $F$, and let $\ell$ be a banal prime with respect to $G$. Then every smooth irreducible $\mathbb{F}_\ell$-representation of $G(F)$ admits a lift to a smooth irreducible $\mathbb{Q}_\ell$-representation.

• Extension: The result is extended to unitary groups and non-split cases (Corollary 6.4), provided the cuspidal support satisfies a "banal" lifting condition.

Technical Lemmas and Intermediate Results

1. Irreducibility of Reductions: If an integral $\mathbb{Q}_\ell$-representation has a banal cuspidal support, its reduction modulo $\ell$ is irreducible (Corollaries 6.1, 6.2, 6.3).
2. Uniqueness of Lifts for Discrete Series: If a representation $\pi \in \text{Irr}_{\mathbb{F}_\ell}(G)$ admits an essentially discrete series lift to $\mathbb{Q}_\ell$, then any lift to $\mathbb{Q}_\ell$ is essentially discrete series (Proposition 6.1).
3. Explicit Construction: The paper provides an explicit recursive algorithm to construct these lifts using derivatives and intertwining operators, ensuring that the "banal" nature of $\ell$ prevents the emergence of non-semisimple behavior or unexpected reducibility.

4. Significance and Applications

Resolution of the Modular Theta Correspondence

The most significant application presented is the proof of the Modular Howe Duality Conjecture in the strongly banal case for symplectic-orthogonal and unitary dual pairs.

• Context: The modular local theta correspondence (studied extensively in [29]) relies on "Hypothesis (H)," which asserts the irreducibility of a specific principal series representation modulo $\ell$.
• Breakthrough: Previous work could not explicitly characterize the primes satisfying Hypothesis (H). By proving that the reduction of the relevant integral representation is irreducible (Proposition 7.1), the author verifies Hypothesis (H) for all strongly banal primes.
• Result: This completes the proof of Howe duality for type I dual pairs in the strongly banal setting, establishing a robust modular local Langlands correspondence for these groups.

Theoretical Impact

• Parity with $GL_n$: The paper demonstrates that for classical groups, the "banal" regime is structurally identical to the complex case, just as it is for $GL_n$. This validates the intuition that large primes behave "nicely" in modular representation theory.
• New Proof for $GL_n$: The methods developed (specifically the use of intertwining operators and derivatives in the modular setting) provide a new, self-contained proof for the lifting theorem in the general linear case, independent of previous specialized techniques.

Summary

Johannes Droschl's paper establishes a comprehensive lifting theorem for classical groups in the banal regime. By synthesizing the theory of intertwining operators, derivatives, and progenerators, the author proves that the representation theory of symplectic, orthogonal, and unitary groups over $\mathbb{F}_\ell$ (for banal $\ell$) is a direct reduction of their $\mathbb{Q}_\ell$ counterparts. This foundational result resolves the final missing step in proving the modular Howe duality conjecture for type I dual pairs, significantly advancing the understanding of the modular Local Langlands program.