Towards the $p$-adic Hodge parameters in semistable representations of $\mathrm{GL}_n(\mathrm{Q}_p)$

Abstract: Let $ρp$ be an $n$-dimensional non-critical semistable $p$-adic Galois representation of the absolute Galois group of $\mathrm{Q}_p$ with regular Hodge--Tate weights. Let $\mathrm{D}$ be the associated $(\varphi,Γ)$-module over the Robba ring. By combining Ding's and Breuil--Ding's methods for the crystalline case with Qian's computation of higher extension groups of locally analytic generalized Steinberg representations, we capture the full information of the $p$-adic Hodge parameters of $ρ_p$ on the automorphic side by considering several Steinberg subquotients of $\mathrm{D}$ and the "crystalline" Hodge parameters between them. These results also admit geometric and Lie-algebraic reformulations on flag varieties related to the moduli space of Hodge parameters. We then construct an explicit locally analytic representation $π{1}(ρp)$ and explicitly describe which Hodge-parameters information of $ρ_p$ it determines. In particular, if the monodromy rank of $ρ_p$ is at most $1$, $π{1}(ρp)$ determines $ρ_p$. When $ρ_p$ comes from a $p$-adic automorphic representation, we show that $π{1}(ρp)$ is a subrepresentation of the $\mathrm{GL}_n(\mathrm{Q}_p)$-representation globally associated to $ρ_p$, under mild hypotheses. Although it is still difficult to construct an explicit representation $π{1}(ρ_p)$ that determines $ρ_p$, our results provide new evidence for the $p$-adic Langlands program in general semistable cases and demonstrate the broad applicability of Ding's, Breuil--Ding's, and Qian's methods.

• The paper introduces automorphic constructions that recover the full Hodge filtration in non-critical semistable p-adic Galois representations.
• It utilizes extension-theoretic techniques and flag variety geometry to parameterize Hodge data via Steinberg blocks and crystalline subquotients.
• Explicit isomorphisms between automorphic extension groups and p-adic Hodge spaces provide strong evidence supporting the p-adic Langlands program.

$p$-adic Hodge Parameters in Semistable Representations of $\mathrm{GL}_n(\mathbb{Q}_p)$

Introduction

This paper addresses the problem of realizing $p$-adic Hodge parameters on the automorphic side for non-critical semistable $p$-adic Galois representations $\rho_p$ of the absolute Galois group of $\mathbb{Q}_p$, with regular Hodge–Tate weights. The work extends the existing theory for crystalline and crystabelline cases to general semistable representations, combining methods from Ding, Breuil–Ding, and Qian to analyze the interaction between $p$-adic Hodge theory and the local $p$-adic Langlands program for $\mathrm{GL}_n(\mathbb{Q}_p)$.

Theoretical Framework and Motivations

Let $\rho_p$ be a de Rham representation of dimension $\mathrm{GL}_n(\mathbb{Q}_p)$0 over a finite extension $\mathrm{GL}_n(\mathbb{Q}_p)$1 of $\mathrm{GL}_n(\mathbb{Q}_p)$2. Fontaine's functoriality attaches to $\mathrm{GL}_n(\mathbb{Q}_p)$3 a filtered $\mathrm{GL}_n(\mathbb{Q}_p)$4-module $\mathrm{GL}_n(\mathbb{Q}_p)$5 over the Robba ring and an irreducible smooth $\mathrm{GL}_n(\mathbb{Q}_p)$6-representation $\mathrm{GL}_n(\mathbb{Q}_p)$7 via the local Langlands correspondence. However, only the $\mathrm{GL}_n(\mathbb{Q}_p)$8-semisimplification of the associated Weil–Deligne representation is captured on the automorphic side by the locally algebraic representation $\mathrm{GL}_n(\mathbb{Q}_p)$9, and crucial finer Hodge-theoretic information---notably the Hodge filtration---is lost.

A central challenge is to recover the Hodge filtration from data on the automorphic side, thus accessing the full moduli space of non-critical $p$0-adic Hodge parameters corresponding to the given monodromy type. The paper systematically resolves this challenge for general semistable non-critical $p$1, identifying new invariants and explicit automorphic constructions that capture these Hodge-theoretic parameters.

Parabolic Filtrations and $p$2-adic Hodge Parameters

By decomposition under the action of Frobenius $p$3 and monodromy $p$4, the $p$5-semisimple structure of $p$6 induces two combinatorial invariants $p$7. $p$8 records the monodromy types, and $p$9 encodes multiplicity relations among $p$0-eigenvalues. The moduli space $p$1 of non-critical $p$2-adic Hodge parameters is identified as a generalized flag variety modulo the action of parabolics determined by $p$3. The paper characterizes "non-criticality" as the relative general position of the Hodge filtration with respect to $p$4-stable flags and formalizes the parameter space as an intersection in the double coset space.

In the two extremal cases, crystalline ($p$5) and full monodromy (Steinberg, $p$6), the parameter space reduces to earlier studied models. The innovation of this work is in the mixed, general semistable scenario: the authors define a sophisticated stratification of the filtered module by Steinberg subquotients and companion crystalline blocks, each associated to intricate extension graphs.

Main Technical Advances

1. Steinberg Blocks and “Crystalline” Parameters

By dissecting $p$7 into admissible compositions of Steinberg blocks and crystalline subquotients, one isolates strata where the Hodge parameters can be determined by extension data between various graded pieces. These subquotients admit $p$8-triangulations, and the Hodge filtration is translated into parameters living in the associated flag varieties and in the moduli of Breuil–Schraen $p$9-invariants.

2. Automorphic Capture via Generalized Steinberg Extensions

The automorphic realization is accomplished by relating the structure of $\rho_p$0-module extensions to extension groups of locally analytic generalized Steinberg representations. The work proves strong isomorphisms and injectivity results between extension groups in the automorphic category and Hodge parameter spaces from $\rho_p$1-adic Hodge theory---for example:

• The parameters of the collection of Steinberg subquotients ("blocks") suffice to determine the $\rho_p$2-adic Hodge parameters in cases where the monodromy has rank at most one;
• In all cases, adding certain "crystalline" parameters or further extension data (between blocks) is sufficient to recover the entire filtration datum.

3. Explicit Construction of Locally Analytic Representations

The authors construct several families of locally analytic representations $\rho_p$3 with explicit branching and socle structures that are shown to contain all the Hodge parameter information for $\rho_p$4 with suitable monodromy type, and in several cases (notably, monodromy rank at most one), to determine $\rho_p$5 itself.

4. Geometric and Lie Algebraic Reformulations

The identification of parameters in flag varieties is further refined to an explicit correspondence between filtered data in Lie algebra coadjoint orbits and automorphic extension groups (via adjoint action and cup-product structures in cohomology). The injectivity and surjectivity theorems connect the geometric moduli space directly to automorphic invariants.

Main Numerical and Algebro-Geometric Results

The authors prove several main theorems:

• For $\rho_p$6 (monodromy type with two blocks), the combinatorial data of the associated Steinberg parameters determines the Hodge filtration.
• For $\rho_p$7, a combination of Steinberg parameters and crystalline subquotient parameters suffices, and this matching is encoded explicitly in morphisms of flag varieties.
• The surjectivity of the composite automorphic realization map is established via intricate dimension counts and commutative diagrams in extension categories.

Moreover, the paper provides explicit recursion and induction on the depth of parabolic filtrations, ensuring that higher-level parameters are either determined by lower-level strata or require only one further crystalline invariant per block.

Implications and Outlook

Practically, these results verify that a large portion of the fine Hodge-theoretic structure of $\rho_p$8---which is essential for understanding the relationships between Galois representations and automorphic representations---is automorphically visible via locally analytic representations that arise naturally in the context of the $\rho_p$9-adic Langlands program. The explicit extension-theoretic classification and construction of representations provide robust evidence for Langlands functoriality in the non-critical, semistable case and clarify the "hidden" structure beyond the classical local Langlands correspondence.

Theoretically, this advances understanding of the relationship between filtered $\mathbb{Q}_p$0-module theory, geometric flag varieties, and analytic representations, suggesting a blueprint for constructing a full $\mathbb{Q}_p$1-adic local Langlands correspondence for $\mathbb{Q}_p$2. It also indicates pathways for generalizing to potentially semistable representations, arithmetic families, and the broader eigenvariety program.

Potential future directions include:

• Extension of the techniques to account for critical Hodge–Tate weights, potentially via non-generic parameters, and to representations of other reductive groups.
• Explicit algorithmic or computational realization of these automorphic correspondences for forms of higher rank and for arithmetic families of representations.
• Exploration of analogous constructions in the relative and global settings, shedding light on explicit local-global compatibility phenomena.

Conclusion

This paper provides a comprehensive parameterization of $\mathbb{Q}_p$3-adic Hodge data for semistable representations of the absolute Galois group of $\mathbb{Q}_p$4, constructing explicit automorphic representations whose structure encodes these parameters and demonstrating the effectiveness of extension-theoretic and flag-geometry techniques in the $\mathbb{Q}_p$5-adic Langlands program. The results offer strong evidence that the $\mathbb{Q}_p$6-adic analytic Langlands correspondence "sees" the entirety of the Hodge filtration for a broad class of non-critical semistable representations, and establish a framework with significant potential for further theoretical and explicit advances in the arithmetic of $\mathbb{Q}_p$7-adic Galois and automorphic representations.