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

TL;DR

This paper demonstrates that the $Cos^lambda$ and $Sin^lambda$ transforms on Grassmann manifolds act as intertwining operators between certain principal series representations of SL(n+1,K), connecting convex geometry with representation theory.

Contribution

It establishes the $Cos^lambda$ and $Sin^lambda$ transforms as standard intertwining operators and analyzes their meromorphic extension and invertibility in the context of representation theory.

Findings

Transforms are meromorphically extendable and invertible for generic parameters.

The K-spectrum of the transforms is explicitly determined.

The work links convex integral geometry with representation theory of semisimple Lie groups.

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 $Sin^\lambda$-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.