Locally Analytic Jacquet–Langlands

• Locally analytic Jacquet–Langlands correspondence is a framework refining classical duality by linking p-adic locally analytic representations with Banach and rigid-analytic geometric structures.
• It leverages period maps, cohomological techniques, and Scholze's patching functors to transfer analytic structures between p-adic groups and their inner forms.
• This correspondence establishes precise links between infinitesimal characters and bounds on the Gelfand–Kirillov dimension in completed cohomology of Shimura varieties.

The locally analytic Jacquet–Langlands correspondence provides a precise framework linking the theory of locally analytic representations of $p$-adic groups, their Banach counterparts, and rigid-analytic geometry associated to local Shimura varieties. This correspondence refines the classical Jacquet–Langlands paradigm, advancing beyond smooth duality and $L$-packets to a “locally analytic” level that captures detailed structures tied to Banach and locally analytic representations, infinitesimal characters, and $p$-adic Hodge-theoretic invariants. The theory relies on modern cohomological and geometric methods—particularly Scholze's patching functors and the theory of diamonds—in conjunction with representation-theoretic and global-automorphic tools.

1. Foundational Notions: Local Shimura Data and Towers

The locally analytic Jacquet–Langlands correspondence is formally realized in the context of local Shimura data $(G,b,\mu)$ defined over $\mathbb{Q}_p$, with a dual datum $(\check G,\check b,\check\mu)$ capturing the inner form correspondence. Associated infinite-level towers $\mathcal{M}_{G,b,\mu,\infty}$ and $$\mathcal{M}_{\check G, \check b, \check\mu, \infty}$$ (in the sense of diamonds) support commuting smooth group actions of both $G=\mathbf{G}(\mathbb{Q}_p)$ and its inner form $G_b$, providing a geometric module for constructing the functorial bridge between their respective representation categories (Dospinescu et al., 2024).

On these towers, one builds period sheaves such as the completed structure sheaf $\widehat{\mathcal{O}}$ as pro-étale (or solid) sheaves. The towers admit compatible period maps (Gross–Hopkins, Hodge–Tate, etc.) into flag varieties, playing a crucial role in transferring representation-theoretic structures across the duality.

The passage from global to local settings is made explicit in the context of Shimura curves attached to quaternion algebras: for instance, letting $D$ be an indefinite quaternion algebra over $\mathbb{Q}$ ramified at $p,\infty$, the tower of Shimura curves $S_{K^p K_p}$ exhibits both $p$-adic uniformization and a transfer between automorphic forms on definite/indefinite forms (Li et al., 20 Jan 2026).

2. Construction of the Locally Analytic Jacquet–Langlands Functor

For $G=\mathrm{GL}_n(F)$ and its basic inner form $D^\times$, the locally analytic Jacquet–Langlands functor $\mathrm{JL}_p$ is defined via cohomology with coefficients in locally analytic sheaves descended via the towers' period maps. Given an admissible locally analytic representation $\pi\in\mathrm{Rep}^{\rm la}(GL_n(F))$, one defines a pro-étale sheaf $\mathcal{F}_\pi$ on $\mathbb{P}^{n-1}$ by Galois descent from the Gross–Hopkins period map. The functor is realized by

$$\mathrm{JL}_p(\pi) = R\Gamma_{\text{proét}}\left(\mathbb{P}^{n-1}_{C_p}, \mathcal{F}_\pi\right).$$

A key theorem asserts that this assignment commutes with the passage to locally analytic vectors: $$\left(\mathrm{JL}_p(\pi)\right)^{D^\times\text{-la}} \cong R\Gamma_{\text{proét}}\left(\mathbb{P}^{n-1}, \mathcal{F}_{\pi^{\rm la}}\right),$$ showing that locally analytic Jacquet–Langlands transfer preserves the analytic structure of representations (Dospinescu et al., 2024).

The construction for $\mathrm{GL}_2$ and $D_p^\times$—which act on the Lubin–Tate and Drinfeld towers—exemplifies this theory, as established by Pan and generalized to arbitrary local Shimura data by Dospinescu, Camargo, and Rodríguez Camargo.

3. Infinitesimal Characters and Gelfand–Kirillov Dimension

A salient feature of the correspondence is compatibility with infinitesimal characters. For irreducible, non-ordinary Banach representations $\Pi$ of $\mathrm{GL}_2(\mathbb{Q}_p)$, Scholze's functor $S_1$ produces a Banach representation of $D^\times$ whose subspace of locally analytic vectors $S_1(\Pi)^{\rm la}$ admits the same infinitesimal character as $\Pi^{\rm la}$. This is realized under the identification of the centers of the enveloping algebras of the respective Lie algebras: $$\mathcal{Z}(\mathfrak{gl}_n) \cong \mathcal{Z}(\mathfrak{d}),$$ ensuring precise transfer of central character data (Dospinescu et al., 2022, Dospinescu et al., 2024).

Moreover, new upper bounds are established for the Gelfand–Kirillov (GK) dimension of admissible Banach representations with an infinitesimal character. Notably, for $\mathrm{GL}_2(\mathbb{Q}_p)$, any such representation $\Pi$ satisfies

$d_{\mathbb{Q}_p}(\Pi^{\rm la}) < 2,$

and in fact $d_{\mathbb{Q}_p}(\Pi^{\rm la}) \leq 1$. This strict inequality leads to finiteness results for irreducible constituents in completed cohomology and for images of the Jacquet–Langlands functor, even bypassing the $p$-adic local Langlands classification in certain cases (Dospinescu et al., 2022).

4. Global Methods and Completed Cohomology

The global realization of the locally analytic correspondence arises in the study of completed cohomology of quaternionic Shimura curves. The completed cohomology $\widehat{H}^1(K^p,E)$, carrying actions of both $G(\mathbb{Q}_p)$ and $D_p^\times$, decomposes into eigenspaces corresponding to Galois representations $$\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(E)$$. For such $\rho$, the locally analytic vectors

$$\check{\Pi}(\rho)^{\rm la} \widehat{\otimes}_E C \cong \tau(\rho_p)^{\oplus m}$$

are described via analytic $D_p^\times$-representations $\tau(\rho_p)$ constructed from the de Rham data of $\rho_p$ and the cohomology of the Lubin–Tate tower. This is analogous to the Breuil–Strauch conjecture for $\mathrm{GL}_2(\mathbb{Q}_p)$, but uniquely captures $D_p^\times$-analytic structure in this context (Li et al., 20 Jan 2026).

Via $p$-adic uniformization (Čerednik–Drinfeld), the analytic sheaf structure on the perfectoid Shimura curve is related to the cohomology of the Lubin–Tate space. Hecke eigenspaces in completed cohomology correspond precisely to locally analytic representations dictated by the functorial Jacquet–Langlands transfer.

5. Cohomological and Functorial Properties

A core result is that the construction of the locally analytic Jacquet–Langlands functor is compatible with derived local analyticity in solid (or condensed) representation theory: $R\Gamma_{\proet}(\mathcal{M}_\infty, \mathcal{B}_I)^{RG\text{-la}} \simeq R\Gamma_{\proet}(\mathcal{M}_\infty, \mathcal{B}_I)^{RG_b\text{-la}}$ for both $G$-side and $G_b$-side analytic vectors. The analytic de Rham cohomology on each finite-level tower and their colimit pass to $G \times G_b$-equivariant isomorphisms, ensuring a strong geometric duality (Dospinescu et al., 2024).

Product formulas for completed cohomology with respect to Hecke eigenspaces and vanishing of higher bi-analytic vectors in the cohomology of towers further guarantee that the correspondence is realized in the setting of locally analytic representations and not lost upon taking analytic vectors.

6. Limitations and Specificities in the Crystalline Case

An important limitation is revealed in the behavior of analytic $D_p^\times$-representations in the crystalline case. When $\rho_p$ is a crystalline Galois representation and its corresponding $\pi_p$ is a principal series, the analytic $D_p^\times$-representation $\tau(\rho_p)$ constructed from the Lubin–Tate tower is independent of the Hodge filtration component of the de Rham Fontaine module $D_{\rm dR}(\rho_p)$. Thus, the locally analytic Jacquet–Langlands functorial image fails to distinguish between two crystalline (potentially non-isomorphic) $\rho_p$ with equal Weil--Deligne parameters but differing Hodge filtrations. In contrast, the locally analytic $\mathrm{GL}_2$-representation $\pi(\rho_p)$, realized via the Drinfeld tower, fully encodes the Hodge filtration (up to twist) (Li et al., 20 Jan 2026).

This subtlety restricts the detection of fine de Rham data to the discrete series case on the quaternionic side, where the extension structure in the analytic representation captures the entirety of the filtration.

7. Overview of Key Results and Their Interrelation

The principal theorems and constructions in the locally analytic Jacquet–Langlands correspondence can be summarized as follows:

Key Object / Result	Description	Source arXiv ID
Locally analytic JL functor $\mathrm{JL}_p$	Maps $$\mathrm{Rep}^{\rm la}(GL_n(F))\to \mathrm{Rep}^{\rm la}(D^\times)$$	(Dospinescu et al., 2024)
Infinitesimal character compatibility	Identifies center actions, preserves analytic structure	(Dospinescu et al., 2022, Dospinescu et al., 2024)
Gelfand–Kirillov dimension bounds	Strict upper bound $d_{\mathbb{Q}_p}(\Pi^{\rm la})<2$ for Banach with inf. char.	(Dospinescu et al., 2022)
De Rham/Lubin–Tate construction	$\tau(\rho_p)$ analytic representation from Lubin–Tate tower	(Li et al., 20 Jan 2026)
Non-detection of Hodge filtration in $\tau(\rho_p)$ (crystalline)	Failure to distinguish crystalline $\rho_p$ differing by filtration	(Li et al., 20 Jan 2026)
Finiteness and explicit local functoriality	Finiteness of irreducible constituents, explicit map $\Pi\mapsto S_1(\Pi)$	(Dospinescu et al., 2022)

The locally analytic Jacquet–Langlands correspondence is thus rigorously constructed at the interface of $p$-adic analytic geometry, representation theory, and arithmetic geometry. It generalizes classical correspondences, encodes intricate analytic and algebraic invariants, and—while subject to limitations in the crystalline, non-discrete series case—provides a powerful duality for locally analytic representations of $p$-adic groups and their inner forms.