Locally Analytic Representations

• Locally analytic representations are p-adic Lie group actions on locally convex spaces defined by convergent p-adic power series, offering a framework that blends functional analysis and nonarchimedean geometry.
• They utilize analytic distribution algebras and solid module theory to establish dualities and compute cohomological invariants via resolutions and explicit complexes.
• These representations have significant applications in p-adic automorphic forms, eigenvarieties, and the p-adic Langlands program, with modern adaptations in condensed mathematics enhancing their study.

A locally analytic representation is a linear action of a $p$-adic Lie group on a locally convex vector space (or, in the solid/condensed framework, a module over an analytic ring) such that the orbit maps are locally given by convergent $p$-adic power series. The modern development of the theory, motivated by $p$-adic Hodge theory, $p$-adic automorphic forms, and the $p$-adic Langlands correspondence, integrates functional analysis, nonarchimedean geometry, and homological algebra. Below, the main definitions, constructions, and categorical frameworks for locally analytic representations are summarized, with a focus on their solid/condensed enhancements and mixed characteristic generalizations.

1. Foundational Definitions and Functional Analytic Framework

Let $G$ be a $p$-adic Lie group (e.g., a compact or locally compact analytic group over a finite extension of $\mathbb{Q}_p$), and let $K$ be a complete nonarchimedean field, often a finite extension of $\mathbb{Q}_p$. A locally analytic representation of $p$0 over $p$1 is a continuous action of $p$2 on a locally convex $p$3-vector space $p$4 such that for every $p$5, the orbit map $p$6 is locally given by convergent $p$7-adic power series around the identity; equivalently, for each $p$8 there is an open compact subgroup $p$9 such that $p$0 is analytic on $p$1 (Jacinto et al., 2021, Jacinto et al., 2023).

The space of locally analytic vectors in $p$2, denoted $p$3, can be characterized as the union over all $p$4 of vectors for which the orbit map is analytic in a neighborhood of radius $p$5; this coincides with the image under the analytic vectors functor defined using the locally analytic distribution algebra $p$6 (Jacinto et al., 2023, Jacinto et al., 2021). Dualities between locally analytic representations on spaces of compact type and coadmissible modules over $p$7, in the sense of Schneider–Teitelbaum, underlie much of the theory (Orlik et al., 2010).

In the framework of condensed mathematics, a "solid" representation is a module over the analytic ring $p$8 (with $p$9 the valuation ring), and the passage to analytic vectors is formulated via adjoints to forgetful functors from module categories over analytic distribution algebras (Jacinto et al., 2023, Jacinto et al., 2021). This language allows for stronger functional-analytic and homological stability.

2. Distribution Algebras, Solid Modules, and Analytic Vectors

The algebra of locally analytic distributions $p$0 is the strong (topological) dual of $p$1, the space of locally analytic $p$2-valued functions on $p$3, and carries a Fréchet–Stein algebra structure, admitting canonical systems of Banach quotient algebras $p$4 indexed by an "analyticity parameter" (Jacinto et al., 2023). The category of solid locally analytic $p$5-representations identifies with the category of quasi-coherent modules over $p$6 (Jacinto et al., 2023).

For a given representation $p$7 in this category, the subspace of analytic vectors with respect to a uniform pro-$p$8 subgroup $p$9 can be captured as the duals of $p$0-modules, and the full functorial exactness is shown for admissible representations (Lahiri, 2019). This identification generalizes classical results about analytic vectors in the context of compact $p$1-adic Lie groups and underpins the interplay between continuous, smooth, and locally analytic representation categories.

3. Resolutions, Homological Methods, and Duality

Projective resolutions and complex-theoretic approaches are central in analyzing locally analytic representations and their extensions. The structure of resolutions adapted from the Schneider–Stuhler complex allows the computation of $p$2 groups for admissible locally analytic representations via explicit complexes built from spaces of analytic vectors and Chevalley–Eilenberg resolutions (Agrawal et al., 2024, Lahiri, 2020).

Bernstein–Zelevinsky and Grothendieck–Serre duality frameworks extend to the locally analytic setting. For locally analytic principal series, Koszul-type resolutions and their duals yield explicit isomorphisms between Ext-complexes, providing a derived category equivalence between the principal series and induced modules from dual Verma (or parabolic Verma) modules, as well as coherent sheaf interpretations on patched eigenvarieties (Strauch et al., 8 Jan 2025). This connects the duality theory for representations with duality on eigenvarieties, crucial for $p$3-adic Langlands and automorphic applications.

4. Geometric and Localization Methods: Buildings, Flag Varieties, and Beyond

The geometric realization of locally analytic representation theory is achieved through localization functors and sheaves on the Bruhat-Tits building or formal models (e.g., for $p$4 over the semistable models of $p$5) (Patel et al., 2014, Patel et al., 2012). For an admissible locally analytic $p$6-representation with prescribed infinitesimal character, there is an equivalence between the coadmissible module category over $p$7 and certain categories of equivariant sheaves over the building. This interpolates between the classical Beilinson–Bernstein localization (on flag varieties for algebraic representations) and the Schneider–Stuhler theory (for smooth representations), with the analytic vector functor linking all three (Schmidt, 2011, Patel et al., 2012).

For $p$8-adic symmetric spaces and moduli spaces (e.g., Lubin–Tate and Drinfeld towers), the robust analytic structures on global sections of equivariant line bundles produce strong duals that are locally analytic $p$9-representations (Sheth, 2017). Local analytic or rigid-analytic cohomology can capture spaces of automorphic forms and their Galois representations, as in completed cohomology and eigenvariety settings (Qiu et al., 15 May 2025).

5. Mixed Characteristic, Solid Theory, and Condensed Mathematics

The "solid" theory, in the sense of Clausen–Scholze's condensed mathematics, upgrades the functional-analytic category to allow for generalized coefficients (Banach pairs of mixed characteristic, such as $G$0, $G$1), and manages actions that are only locally analytic in a more flexible, robust way (Porat, 15 Oct 2025). The resulting module categories admit tensor products, derived functors, full six-functor formalisms, and powerful homological comparison theorems (e.g., between continuous, analytic, and Lie algebra cohomology) (Jacinto et al., 2023, Jacinto et al., 2021). This approach permits the systematic study of analytic $G$2-modules, extended eigenvarieties, and $G$3-adic Langlands correspondences in both characteristic zero and $G$4.

The idempotency and cohomological comparison theorems about analytic distribution algebras, the "decompletion" results underlying Sen theory and $G$5-adic Hodge decompositions, as well as eigenvariety constructions with extended coefficients, all become accessible within this enhanced categorical setting (Porat, 15 Oct 2025).

6. Parabolic Induction and Non-Principal Series Constructions

Locally analytic representations are not limited to principal series or their smooth analogues. Induction from Lie algebra modules outside the classical category $G$6, including Whittaker and cuspidal modules, produces genuinely non-principal and often supercuspidal (in the homological sense) representations (Orlik, 29 Oct 2025, Orlik, 7 May 2025). These constructions, though often yielding inadmissible or only ind-admissible objects, exhibit new homological and irreducibility properties—such as vanishing Jacquet modules and homological vanishing conditions for all parabolic subgroups—providing counterexamples and testing grounds for the structure theory of $G$7-adic groups.

Translation functors, as defined for coadmissible $G$8-modules, mirror the algebraic category $G$9 translation functors and are realized on the analytic side with compatibilities to the $p$0-adic Langlands correspondence (Jena et al., 2021).

7. Applications and Further Directions

Locally analytic representation theory underpins $p$1-adic automorphic forms, the $p$2-adic local Langlands program, the construction and study of eigenvarieties, and the analysis of completed (and locally analytic) cohomology of Shimura and Rapoport–Zink spaces (Dospinescu et al., 2024, Qiu et al., 15 May 2025). Integration with condensed mathematics extends the reach of the theory to broader coefficient systems and to categorical and geometric $p$3-adic Langlands correspondences. Several open problems remain, including the extension to non-compact groups, refinements to include more general coefficients and compactifications, the intrinsic construction of mixed-characteristic differential operators, and the explicit computation of Ext groups and dimensions for general modules (Porat, 15 Oct 2025). The solid and geometric approaches promise to play a central role in future developments of nonarchimedean representation theory.