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