Repr\'esentations des quaternions de norme 1

TL;DR

This paper classifies smooth irreducible representations of the norm 1 elements in a quaternion division algebra over a local field, extending previous work on SL_2(F) and analyzing restrictions based on characteristic and dimension.

Contribution

It provides a classification of irreducible representations of D^1 and describes their restrictions, generalizing earlier results for SL_2(F) to quaternion division algebras.

Findings

Restriction of representations is irreducible or sum of two irreducibles when dimension > 1.

Restriction is sum of two equivalent irreducibles iff certain conditions hold in the Jacquet-Langlands correspondence.

In characteristic not p, the sum of four inequivalent irreducibles occurs only under specific conditions, never if characteristic is 2.

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 D^1 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 SL_2(F), we show that the restriction of V* to D^1 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 GL_2(F) image of V* by the Jacquet-Langlands correspondence restricts to SL_2(F) as a sum of four inequivalent irreducible representations (this is never the case if the characteristic of R is not 2).