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On Poisson transforms for spinors

Salem Ben Saïd, Abdelhamid Boussejra and Khalid Koufany

Vol. 5 (2023), No. 4, 771–792
Abstract

Let (τ,V τ) be a spinor representation of Spin (n) and let (σ,V σ) be a spinor representation of Spin (n 1) that occurs in the restriction τ| Spin (n1).

We consider the real hyperbolic space Hn() as the rank one symmetric space Spin 0(1,n)Spin (n) and the spinor bundle ΣHn() over Hn() as the homogeneous bundle Spin 0(1,n) ×Spin (n)V τ.

In this paper we characterize eigenspinors of the algebra of invariant differential operators acting on ΣHn() which can be written as the Poisson transform of Lp-sections of the bundle Spin (n) ×Spin (n1)V σ over the boundary Sn1 Spin (n)Spin (n 1) of Hn(), for 1 < p < .

Keywords
Poisson transform, real hyperbolic space, Spin representation, Spinor bundle
Mathematical Subject Classification
Primary: 15A66, 43A85, 22E30
Milestones
Received: 6 December 2022
Revised: 22 July 2023
Accepted: 7 August 2023
Published: 21 November 2023
Authors
Salem Ben Saïd
Department of Mathematical Sciences, College of Science
United Arab Emirates University
Abu Dhabi
United Arab Emirates
Abdelhamid Boussejra
Department of Mathematics, Faculty of Sciences
University Ibn Tofail
Kenitra
Morocco
Khalid Koufany
Institut Elie Cartan de Lorraine
Université de Lorraine
Vandoeuvre-Les-Nancy
France