Links between generalized Montréal-functors

• Published in: Mathematische Zeitschrift 2017-08, Vol.286 (3-4), p.1227-1275, Article 1227
• Main Authors: Erdélyi, Márton, Zábrádi, Gergely
• Format: Article
• Language: English
• Subjects: Algebra; Clay; Integers; Links; Mathematics; Mathematics and Statistics; Number theory; Torsion
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-016-1799-2

Let o be the ring of integers in a finite extension K / Q p and G = G ( Q p ) be the Q p -points of a Q p -split reductive group G defined over Z p with connected centre and split Borel B = TN . We show that Breuil’s (Algebra Number Theory 9(10):2241–2291, 2015 ) pseudocompact ( φ , Γ ) -module D ξ ∨ ( π ) attached to a smooth o -torsion representation π of B = B ( Q p ) is isomorphic to the pseudocompact completion of the basechange O E ⊗ Λ ( N 0 ) , ℓ D S V ~ ( π ) to Fontaine’s ring (via a Whittaker functional ℓ : N 0 = N ( Z p ) → Z p ) of the étale hull D S V ~ ( π ) of D S V ( π ) defined by Schneider and Vigneras (Clay Math Proc 13:525–601, 2011 ). Moreover, we construct a G -equivariant map from the Pontryagin dual π ∨ to the global sections Y ( G / B ) of the G -equivariant sheaf Y on G  /  B attached to a noncommutative multivariable version D ξ , ℓ , ∞ ∨ ( π ) of Breuil’s D ξ ∨ ( π ) whenever π comes as the restriction to B of a smooth, admissible representation of G of finite length.