Cuspidal $\ell$-modular representations of ${\rm GL}_n(F)$ distinguished by a Galois involution, II

Abstract: Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\neq2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of ${\rm GL}_n(F)$ which are ${\rm GL}_n(F_0)$-distinguished, that is, which carry a non-zero ${\rm GL}_n(F_0)$-invariant linear form. In the case when $\ell\neq2$, an $\ell$-modular cuspidal representation of ${\rm GL}_n(F)$ is ${\rm GL}_n(F_0)$-distinguished if and only if it lifts to a ${\rm GL}_n(F_0)$-distinguished cuspidal $\ell$-adic representation, whereas when $\ell=2$, it is ${\rm GL}_n(F_0)$-distinguished if and only if it is conjugate-self-dual.

• The paper classifies H-distinguished cuspidal ℓ-modular representations of GL_n(F) by establishing an equivalence with the existence of distinguished lifts when ℓ≠2.
• It employs modular Rankin–Selberg gamma factors and type theory to analyze supercuspidal support and arithmetic invariants governing distinction.
• For ℓ=2, the study finds that distinction is determined solely by o-self-duality, highlighting novel modular phenomena in p-adic harmonic analysis.

Classification of Cuspidal $\ell$-Modular Representations of ${\rm GL}_n(F)$ Distinguished by a Galois Involution

Introduction and Background

The paper studies the problem of classifying irreducible cuspidal $\ell$-modular representations of $G = {\rm GL}_n(F)$ that are distinguished by the fixed points $H = {\rm GL}_n(F_0)$ under a Galois involution, where $F/F_0$ is a quadratic extension of non-Archimedean local fields with residual characteristic $p \neq 2$ and $\ell \neq p$. The key notion is $H$-distinction: a representation is called $H$-distinguished if it admits a nontrivial ${\rm GL}_n(F)$0-invariant linear form. The theory is highly sensitive to the characteristic of the field of coefficients and the residue characteristic, as well as the parity of the prime ${\rm GL}_n(F)$1.

The context connects to the harmonic analysis on ${\rm GL}_n(F)$2-adic groups, distinction and periods, and the local Langlands correspondence, including analogues and applications in the modular representation theory (i.e., in characteristic ${\rm GL}_n(F)$3 rather than characteristic zero). In complex coefficients, the distinction problem is controlled by conjugate-self-duality and more refined local parameter properties (conjugate-orthogonality), but modular scenarios are more subtle due to new phenomena such as failure of the dichotomy and complications with lifting and reduction modulo ${\rm GL}_n(F)$4.

Main Results and Classification Statements

The Case ${\rm GL}_n(F)$5

A principal result is a complete classification of ${\rm GL}_n(F)$6-distinguished cuspidal irreducible ${\rm GL}_n(F)$7-representations of ${\rm GL}_n(F)$8 for ${\rm GL}_n(F)$9.

Main Theorem (Theorem 1.2 / 3.1): A cuspidal $\ell$0-representation of $\ell$1 is $\ell$2-distinguished if and only if it has a $\ell$3-distinguished cuspidal $\ell$4-lift.

This result achieves an explicit linkage between modular distinction and the existence of distinguished lifts in the $\ell$5-adic category. The technical core involves an analysis of supercuspidal support, use of gamma factors (specifically, modular Rankin–Selberg local factors), and a detailed study of the invariants governing distinctions and lifts. The key insight is that, for $\ell$6, the obstruction to distinction at the modular level aligns with the absence of a distinguished lift.

Consequences in $\ell$7 Case

• Disjunction: A cuspidal representation cannot be both $\ell$8- and $\ell$9-distinguished (where $G = {\rm GL}_n(F)$0 is the character with kernel the group of norms), reflecting a modular analogue of the classical dichotomy (Corollary 3.19), but the dichotomy fails in general for modular non-supercuspidal cuspidal representations.
• The classification itself (Theorems 3.8, 3.9) is given in terms of explicit type-theoretic and arithmetic invariants attached to the supercuspidal support, unramified base field, and behavior under Galois.
• The method develops and exploits fine results on modular gamma factors over finite fields and utilizes strong lifting theorems from type theory and Bushnell-Kutzko machinery.

The Case $G = {\rm GL}_n(F)$1

In stark contrast, the case $G = {\rm GL}_n(F)$2 yields a remarkable simplification.

Main Theorem (Theorem 1.3 / 7.1): A cuspidal $G = {\rm GL}_n(F)$3-representation of $G = {\rm GL}_n(F)$4 is $G = {\rm GL}_n(F)$5-distinguished if and only if it is $G = {\rm GL}_n(F)$6-self-dual (that is, isomorphic to its contragredient conjugated by the nontrivial Galois automorphism of $G = {\rm GL}_n(F)$7).

This result extends the self-duality criterion—previously valid for supercuspidal representations—to all cuspidal irreducibles in characteristic $G = {\rm GL}_n(F)$8. It highlights an exceptional behavior at $G = {\rm GL}_n(F)$9: the dichotomy with $H = {\rm GL}_n(F_0)$0-distinction disappears, and the sole obstruction for modular distinction is conjugate self-duality. The proof leverages a detailed analysis of the structure of induced representations, explicit Mackey theory, and a contradiction argument via the dimension of invariants.

Technical Innovations

• The core arguments are based on comparing two independent calculations of modular Rankin-Selberg $H = {\rm GL}_n(F_0)$1-factors, under the assumption that a distinguished representation does not admit a distinguished lift. Surprising sign discrepancies in the two evaluations, only possible when $H = {\rm GL}_n(F_0)$2, create the required contradiction.
• The paper introduces a new class $H = {\rm GL}_n(F_0)$3 of modular representations that enables the extension of Ok’s approach on local factors to the modular setting. This is crucial for the finite field case and the analysis of invariant forms.
• Through a careful analysis of “generic” representations, the authors trace how distinction properties and modular gamma factors are preserved or altered under reduction and lifting, and parabolic induction.

Implications and Theoretical Significance

The main results supply a complete modular classification for distinguished cuspidal representations with respect to a Galois involution in $H = {\rm GL}_n(F_0)$4, resolving a long-standing issue in the non-semisimple, modular setting. These theorems demonstrate that, for almost all primes $H = {\rm GL}_n(F_0)$5, modular distinction is governed entirely by the capacity to lift to distinguished representations in characteristic zero, while for $H = {\rm GL}_n(F_0)$6, a pure self-duality criterion emerges universally.

Additionally, the work suggests the following:

• There is no modular dichotomy in general; new behaviors arise that do not mirror the complex (characteristic zero) case.
• The sharp distinction between the cases $H = {\rm GL}_n(F_0)$7 and $H = {\rm GL}_n(F_0)$8 indicates profound subtleties in modular harmonic analysis, suggesting further arithmetic phenomena in the presence of torsion.
• The analysis portends potential generalizations to other involutions, inner forms, and potentially other reductive $H = {\rm GL}_n(F_0)$9-adic groups, although some obstructions already arise in groups beyond $F/F_0$0, as discussed in the paper’s speculative remarks.

Future Directions

• Generalization to arbitrary involutions or inner forms of $F/F_0$1 seems plausible, and ongoing work suggests this classification can be extended, potentially with further group-specific obstructions.
• Understanding whether similar “mod $F/F_0$2 shadow” phenomena for distinction and gamma factors appear in period problems for other $F/F_0$3-adic or finite groups (unitary, symplectic, etc.).
• The detailed study of modular analogues of the Rankin–Selberg or Godement–Jacquet factors for distinctions and their potential connection to the local Langlands correspondence in mod $F/F_0$4 scenarios.
• Investigating possible applications in the context of congruences of automorphic forms, mod $F/F_0$5 representation theory of other groups, and the relative Langlands program.

Conclusion

This work resolves the distinction problem for $F/F_0$6-modular cuspidal representations of $F/F_0$7 with respect to a Galois involution: for $F/F_0$8, distinction reflects the existence of a distinguished lift; for $F/F_0$9, distinction coincides with $p \neq 2$0-self-duality. The approach combines explicit harmonic and representation-theoretic techniques with delicate analysis of modular local factors. These results clarify and complete the understanding of modular distinguished representation theory for general linear groups, providing a foundation for further developments in modular harmonic analysis, relative Langlands theory, and the study of periods and distinction in torsion settings.