The Cos λ and Sin λ transforms as intertwining operators between generalized principal series representations of SL(n + 1, K)

Gestur Olafsson (1) , Angela Pasquale (2)

1 LSU - Louisiana State University
2 LMAM - Laboratoire de Mathématiques et Applications de Metz

In this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple Lie groups by showing that the Cos λ and Sin λ-transforms on the Grassmann manifolds Gr p (K) = SU(n + 1, K)/S(U(p, K) × U(n + 1 − p, K)) are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup P p of SL(n + 1, K). The index p indicates the dependence of the parabolic on p. The general results of Knapp and Stein and Vogan and Wallach then show that both transforms have meromorphic extension to C and are invertible for generic λ ∈ C. Furthermore, known methods from representation theory combined with a Selberg type integral allow us to determine the K-spectrum of those operators.

Mots clés: Representations of semisimple Lie groups; Cosλ and Sinλ transforms; Grassmannian manifolds; Intertwining operators

Domaines: Mathématiques [math]

Gestur Olafsson, Angela Pasquale. The Cos λ and Sin λ transforms as intertwining operators between generalized principal series representations of SL(n + 1, K). Advances in Mathematics, 2012, 229 (1), pp.267-293. ⟨10.1016/j.aim.2011.08.015⟩. ⟨hal-01279429⟩