p-adic Banach space representations of SL₂(\mathbb Q_p)

2019/12/23 by Dubravka Ban, Ban, Dubravka, Matthias Strauch +1
Mathematics · #11S37 #20G25 #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1912.11125

openalex publication_date 2019/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the restriction to SL₂(\mathbb Q_p) of an irreducible p-adic unitary Banach space representation Π of GL₂(\mathbb Q_p). If Π is associated, via the p-adic local Langlands correspondence, to an absolutely irreducible 2-dimensional Galois representation ψ, then the restriction of Π decomposes as a direct sum of r ≤ 2 irreducible representations. The main result of this paper is that r is equal to the cardinality s of the centralizer in PGL₂ of the projective Galois representation ψ associated to ψ, and the restriction is multiplicity-free, except if ψ is triply-imprimitive, in which case the restriction of Π is a direct sum of two equivalent representations. From this result we derive a classification of absolutely irreducible p-adic unitary Banach space representations of SL₂(\mathbb Q_p).

Related

• The p-adic local Langlands correspondence for GL₂(Q_p)
• Représentations cristallines irréductibles de \rm GL₂(Q_p)
• Banach-Hecke algebras and p-adic Galois Representations
• The image of Colmez's Montreal functor
• Locally analytic vectors of unitary principal series of GL₂(Qp)