[Paper Review] Locally analytic vectors of unitary principal series of GL_2(Qp)

Ruochuan Liu, Bingyong Xie+1 more|arXiv (Cornell University)|Mar 13, 2011

Advanced Algebra and Geometry19 references14 citations

TL;DR

This paper proves Emerton's conjecture on the structure of locally analytic vectors in unitary principal series representations of GL₂(ℚₚ) for p > 2. Using Colmez’s p-adic local Langlands machinery, the authors identify the locally analytic vectors as fitting into an exact sequence involving principal series representations and rank-one (φ,Γ)-modules, resolving a key open problem in the p-adic Langlands program for GL₂(ℚₚ).

ABSTRACT

The p-adic local Langlands correspondence for GL2(Qp) attaches to any 2-dimensional irreducible p-adic representation V of the absolute Galois groups of Qp an admissible unitary representation Pi(V) of GL2(Qp). The unitary principal series of GL2(Qp) are those Pi(V) corresponding to trianguline representations. In this article, for p>2, using the machinery of Colmez, we determine the space of locally analytic vectors for all non-exceptional unitary principal series of GL2(Qp) by proving a conjecture of Emerton.

Motivation & Objective

• To resolve Emerton's conjecture on the structure of locally analytic vectors in unitary principal series representations of GL₂(ℚₚ).
• To extend Colmez’s p-adic local Langlands correspondence to the setting of locally analytic vectors for non-exceptional trianguline representations.
• To determine the precise structure of Π(V)ₐₙ for unitary principal series via duality and (φ,Γ)-module theory.
• To establish a precise exact sequence describing the locally analytic vectors in terms of principal series and rank-one modules.

Proposed method

• Utilizes Colmez’s duality formula: (Π(∨V)ₐₙ)* ≅ Dₙₐₜᵣᵢg(∨V) ⊗ₚ¹, linking locally analytic vectors to étale (φ,Γ)-modules.
• Constructs twisted tensor products R(η) ⊗₆ P¹ and R⁺(η) ⊗₆ P¹ to model duals of locally analytic principal series representations.
• Applies the theory of rank-one (φ,Γ)-modules and their extensions to analyze the structure of Dₙₐₜᵣᵢg(s) ⊗ₚ¹ for trianguline representations.
• Uses duality and orthogonality in the P¹-dual space to identify subrepresentations and compute codimensions in exact sequences.
• Applies a key result from Colmez’s theory to relate the image of the natural map jₚ¹ to subspaces of R⁺(δ) ⊗ₚ¹.
• Employs a case analysis based on whether δ₁δ₂⁻¹ = xᵏ|x| for k ∈ ℤ₊ to distinguish between discrete series and principal series structures.

Experimental results

Research questions

• RQ1 What is the precise structure of the space of locally analytic vectors Π(V)ₐₙ for non-exceptional unitary principal series of GL₂(ℚₚ)?
• RQ2 Does Emerton’s conjectured exact sequence 0 → Σ(s) → Π(V(s))ₐₙ → (Ind_B GL₂(ℚₚ) δ₁⊗δ₂(x|x|)⁻¹)ₐₙ → 0 hold for all non-exceptional s ∈ ℒ_irr?
• RQ3 How do the locally analytic vectors relate to the duals of rank-one (φ,Γ)-modules and their extensions?
• RQ4 What is the codimension of the locally analytic subrepresentation inside the full locally analytic vector space in the case δ₁δ₂⁻¹ = xᵏ|x|?
• RQ5 Can Breuil’s more precise conjecture on the structure of Π(V)ₐₙ be deduced from Emerton’s conjecture in the non-exceptional case?

Key findings

• Emerton’s conjecture is proven to be true for all non-exceptional unitary principal series of GL₂(ℚₚ) when p > 2.
• The space of locally analytic vectors Π(V(s))ₐₙ fits into an exact sequence 0 → Σ(s) → Π(V(s))ₐₙ → (Ind_B GL₂(ℚₚ) δ₁⊗δ₂(x|x|)⁻¹)ₐₙ → 0, where Σ(s) is defined via Breuil’s locally analytic representations.
• When δ₁δ₂⁻¹ = xᵏ|x| for some k ∈ ℤ₊, the representation Σ(s) is isomorphic to Σ(k+1, ℒ) ⊗ (δ₂|x|^(2−k)/²∘det), and the quotient has a nontrivial extension structure.
• In the case δ₁δ₂⁻¹ = xᵏ|x|, the dual of the quotient Π(V(s))ₐₙ / Σ(δ₁,δ₂) is isomorphic to R(δ₂) ⊗ₚ¹ ∩ Dₙₐₜᵣᵢg(∨s) ⊗ₚ¹, which is an extension of R⁺(δ₂) ⊗ₚ¹ by a k-dimensional space.
• The dual of the locally analytic representation Σ(s) is isomorphic to R⁺(δ₁) ⊗ₚ¹ when δ₁δ₂⁻¹ ≠ xᵏ|x|, and to an extension of R⁺(δ₁) ⊗ₚ¹ by a k-dimensional space when δ₁δ₂⁻¹ = xᵏ|x|.
• The result implies that Breuil’s conjecture on the structure of Π(V)ₐₙ for non-exceptional representations follows from Emerton’s conjecture, thus generalizing earlier results of the first author.