[Paper Review] Infinitesimal characters in arithmetic families

Gabriel Dospinescu, Vytautas Paškūnas+1 more|arXiv (Cornell University)|Dec 2, 2020

Advanced Algebra and Geometry38 references4 citations

TL;DR

This paper introduces a p-adic analogue of infinitesimal characters for families of L- and C-parameters of p-adic reductive groups, using the Sen operator and Harish-Chandra homomorphism. It proves that locally analytic vectors in completed cohomology of arithmetic groups carry infinitesimal characters compatible with Galois representations, establishing a p-adic Langlands correspondence framework via Hodge–Tate data and analytic continuation.

ABSTRACT

We associate infinitesimal characters to (twisted) families of $L$-parameters and $C$-parameters of $p$-adic reductive groups. We use the construction to study the action of the centre of the universal enveloping algebra on the locally analytic vectors in the Hecke eigenspaces in the completed cohomology.

Motivation & Objective

• To define infinitesimal characters for p-adic families of L- and C-parameters using the Sen operator and Harish-Chandra homomorphism.
• To establish that locally analytic vectors in Hecke eigenspaces of completed cohomology carry infinitesimal characters when compatible with Galois data.
• To show that infinitesimal characters propagate via analytic continuation from classical points to non-classical points in families.
• To connect the infinitesimal character to generalized Hodge–Tate weights of associated Galois representations in arithmetic families.
• To provide a framework for the p-adic Langlands correspondence by relating infinitesimal characters to automorphic and Galois data.

Proposed method

• Constructs infinitesimal characters via the conjugacy class of the semisimple part of the Sen operator acting on Galois representations with values in L- and C-groups.
• Applies Chevalley’s restriction theorem and the Harish-Chandra homomorphism to define infinitesimal characters in the p-adic setting.
• Uses analytic continuation of locally analytic vectors in Banach representations to glue compatible infinitesimal characters across families.
• Relies on the topology of coefficient rings and the structure of Cohen–Macaulay modules to ensure density of algebraic vectors.
• Applies the patching method to completed cohomology of Shimura varieties and locally symmetric spaces to extend results to global settings.
• Establishes compatibility between Hodge–Tate weights and infinitesimal characters under weakly non-Eisenstein conditions.

Experimental results

Research questions

• RQ1 Can infinitesimal characters be defined for p-adic families of L- and C-parameters using Galois-theoretic data such as the Sen operator?
• RQ2 Does the action of the center of the universal enveloping algebra on locally analytic vectors in completed cohomology extend to a well-defined infinitesimal character in families?
• RQ3 To what extent do infinitesimal characters in the p-adic setting reflect the Hodge–Tate weights of associated Galois representations?
• RQ4 Can the infinitesimal character of a classical automorphic form be analytically continued to non-classical points in the eigenvariety?
• RQ5 Is there a canonical infinitesimal character associated to a given C-algebraic automorphic form via a Galois representation?

Key findings

• Infinitesimal characters are constructed for (twisted) families of L- and C-parameters using the semisimple part of the Sen operator, generalizing the classical real case.
• The action of the center of the universal enveloping algebra on locally analytic vectors in completed cohomology is shown to factor through a compatible family of infinitesimal characters.
• In favorable settings—such as completed cohomology of modular curves, Shimura curves, and definite unitary groups—locally analytic vectors carry infinitesimal characters computable from Hodge–Tate data.
• The infinitesimal character at classical points matches the Hodge–Tate weights of the associated Galois representation, supporting a weak form of local-global compatibility at p.
• The results extend to patched modules in the Caleary–Geraghty–Scholze framework, confirming that candidate p-adic local Langlands correspondences have well-defined infinitesimal characters.
• A conjectural Galois representation is shown to induce a well-defined infinitesimal character if it satisfies conditions on Frobenius conjugacy and cyclotomic twist, under the assumption of existence.