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

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.