The $\ct$ transform on line bundles over compact Hermitian symmetric spaces

TL;DR

This paper extends the spectrum analysis of the $Cos^\lambda$ transform from functions to sections of line bundles over Grassmannians, revealing its role as an intertwining operator in representation theory.

Contribution

It introduces the $Cos^\lambda$ transform on line bundles over Grassmannians and determines its spectrum using representation-theoretic methods.

Findings

The $Cos^\lambda$ transform acts as an intertwining operator between principal series representations.

The spectrum of the $Cos^\lambda$ transform is explicitly computed.

The method generalizes previous results from functions to line bundle sections.

Abstract

In a previous article the second author together with A. Pasquale determined the spectrum of the $Cos^\lambda$ transform on smooth functions on the Grassmann manifolds $G_{p,n+1}$. This article extends those results to line bundles over certain Grassmannians. In particular we define the $Cos^\lambda$ transform on smooth sections of homogeneous line bundles over$G_{p,n+1}$ and show that it is an intertwining operator between generalized ($\chi$-spherical) principal series representations induced from a maximal parabolic subgroup of $\mathrm{SL} (n+1, \mathbb{K})$. Then we use the spectrum generating method to determine the $K$-spectrum of the $Cos^\lambda$ transform.