On Poisson transforms for spinors

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( n − 1) . We consider the real hyperbolic space H n ( R ) as the rank one homogeneous space Spin 0 (1 , n ) / Spin( n ) and the spinor bundle Σ H n ( R ) over H n ( R ) as the homogeneous bundle Spin 0 (1 , n ) × Spin( n ) V τ . Our aim is to characterize eigenspinors of the algebra of invariant differential operators acting on Σ H n ( R ) which can be written as the Poisson transform of L p -sections of the bundle Spin( n ) × Spin( n − 1) V σ over the boundary S n − 1 ≃ Spin( n ) / Spin( n − 1) of H n ( R ).