Extension of Symmetric Spaces and Restriction of Weyl Groups and Invariant Polynomials

TL;DR

This paper introduces a criterion for the surjectivity of restricting Weyl group invariant polynomials to subspaces, with applications to Fourier analysis on symmetric spaces.

Contribution

It provides a new simple criterion for the restriction of invariant polynomials, advancing the understanding of polynomial invariants in symmetric space analysis.

Findings

Criterion for surjectivity of polynomial restriction

Application to Fourier analysis on symmetric spaces

Foundation for future Paley--Wiener theorems

Abstract

Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion that ensure that the restriction of invariant polynomials to subspaces is surjective. In another paper we will apply our criterion to problems in Fourier analysis on projective/injective limits, specifically to theorems of Paley--Wiener type.