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

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

2011-03-23

arxiv.org/abs/1103.4557v1

Mathematics

Representation Theory

Scientific paper

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 $Sin^\lambda$-transforms on the Grassmann manifolds $Gr_p(K)=SU (n+1,K)/S (U (p,K)\times U (n+1-p,K))$ are standard 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.

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