The $Co{s}^{\lambda }$ and $Si{n}^{\lambda }$ Transforms as Intertwining Operators between generalized principal series Representations of SL (n+1,K)

Abstract

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 $Cos^\lamda$ and $Si{n}^{\lambda }$-transforms on the Grassmann manifolds $G{r}_p(K)=SU(n+1,K)/S(U(p,K)\times U(n+1-p,K))$ 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 $\lambda \in C$. Furthermore, known methods from representation theory combined with a Selberg type integral allow us to determine the K-spectrum of those operators.