Représentations des quaternions de norme 1

2025/02/20 by Guy Henniart, Henniart, Guy, Marie‐France Vignéras +1
Engineering · Mathematics · #11F70 #22E50 #Advanced Numerical Analysis Techniques #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2502.14346

openalex publication_date 2025/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F be a local field with finite residue field of characteristic p, D the quaternion division algebra with centre F, and R an algebraically closed field of any characteristic. We classify the smooth irreducible R-representations V of the group D¹ of elements of D* with reduced norm 1. Such a V occurs in the restriction of a smooth irreducible R-representation V* of D*. When the dimension of V*is >1, following our previous work in the case of SL₂(F), we show that the restriction of V* to D¹ is irreducible or the sum of two irreducible representations. When the characteristic of R is not p, that restriction is the sum of two irreducible equivalent representations if and only if the representation of GL₂(F) image of V* by the Jacquet-Langlands correspondence restricts to SL₂(F) as a sum of four inequivalent irreducible representations (this is never the case if the characteristic of R is not 2).

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