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In this article we connect topics from convex and integral geometry with wellknown topics in representation theory of semisimple Lie groups by showing thatthe $Cos^\lamda$ and $Sin^\lambda$-transforms on the Grassmann manifolds$Gr_p(K)=SU (n+1,K)/S (U (p,K)\times U (n+1-p,K))$ are standard intertwiningoperators between certain generalized principal series representations inducedfrom 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 Knappand Stein and Vogan and Wallach then show that both transforms have meromorphicextension to C and are invertible for generic $\lambda\in C$. Furthermore,known methods from representation theory combined with a Selberg type integralallow us to determine the K-spectrum of those operators.

The $Cos^\lambda$ and $Sin^\lambda$ Transforms as Intertwining Operators between generalized principal series Representations of SL (n+1,K)