Infinitesimal characters in arithmetic families

Abstract

We associate infinitesimal characters to (twisted) families of Lparameters 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.

On local Galois deformation rings

“…It follows from (18) that 𝑋 gen \ 𝑉 Kirr = 𝑌 ∪ 𝑍 Kred ∪ P min <P 𝑋 gen P . Thus, it follows from (23) and the definition of…”

“…Using the theory of highest weight, we may find 𝜏 ∈ Irr(𝐺) such that the central character of 𝜏 is equal to 𝜃. It follows from [23,Corollary 7.8] Let 𝔞 be the 𝑅 ∞ annihilator of 𝑀 ∞ . In [25, Theorem 6.12], it is shown, following the approach of Chenevier [17] and Nakamura [40], that the closure in Spec 𝑅 ∞ of the union of the supports of 𝑀 ∞ (𝜉 ) for all 𝜉 ∈ Irr(𝐺) is a union of irreducible components of Spec 𝑅 ∞ .…”

“…For each 𝜉 ∈ Irr(𝐺), let 𝔞 𝜉 be the 𝑅 𝜌 -annihilator of 𝑀 ∞ (𝜉). It follows from [14,Lemma 4.18] that 𝑅 𝜌 /𝔞 𝜉 is a quotient of the crystalline deformation ring of 𝜌 with Hodge-Tate weights corresponding to the highest weight of 𝜉; see [14, Section 1.8], [23,Remark 5.14]. Moreover, 𝑅 𝜌 /𝔞 𝜉 is a union of irreducible components of that ring.…”

Local models for the trianguline variety and partially classical families

“…But a stronger version was proved in [18]: in regular cases, (ρ, δ ′ ) exists on Y (U p , ρ) for any refinement R ′ of ρ and δ ′ ∈ W R ′ (ρ). This stronger result is easy to get from Theorem 1.1 in regular cases using locally algebraic vectors in Π(ρ) and is not available in this paper for general crystalline points due to the non-existence of non-zero locally algebraic vectors in Π(ρ) when ρ is non-regular (the non-existence can be seen using the results on infinitesimal characters in [32]). See Remark 5.24 for a partial result.…”

“…Remark 5.17. The assumption that h is regular is necessary: if ρ p is de Rham with non-regular Hodge-Tate weights, Q p = G p and J = Σ p , then there exists no non-zero locally algebraic vector in Π ∞ [m rx ] an by the local-global compatibility or [32].…”