The Cosλ and Sinλ Transforms as Intertwining Operators between generalized principal series Representations of SL (n+1,K)

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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 and Sinλ-transforms on the Grassmann manifolds Grₚ(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 Pₚ 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.