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

摘要

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 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 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 (1971, 1980) [20,21] and Vogan and Wallach (1990) [44] 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.