# Spectral expansions of random sections of homogeneous vector bundles

## TL;DR

This paper develops a unified mathematical framework using group representation theory to rigorously construct isotropic random sections of homogeneous spin and tensor bundles, with applications in astrophysics.

## Contribution

It introduces a comprehensive approach based on induced group representations for constructing isotropic random sections, unifying orthogonal and unitary cases.

## Key findings

- Provides a rigorous mathematical construction method.
- Includes astrophysical examples illustrating the approach.
- Unifies different representation theories in the context of random sections.

## Abstract

Tiny fluctuations of the Cosmic Microwave Background as well as various observable quantities obtained by spin raising and spin lowering of the effective gravitational lensing potential of distant galaxies and galaxy clusters, are described mathematically as isotropic random sections of homogeneous spin and tensor bundles. We consider the three existing approaches to rigourous constructing of the above objects, emphasising an approach based on the theory of induced group representations. Both orthogonal and unitary representations are treated in a unified manner. Several examples from astrophysics are included.