Invariants for the isotropy representation of a Riemannian symmetric space

Question

Statement: Let $M = G/K$ be a Riemannian symmetric space of compact type, and $V = T_o M$ be its isotropy representation (of $K$ acting on the tangent space of $M$). Then the Hilbert–Poincaré series $\sum_{n=0}^{+\infty} \dim(\mathcal{S}^n V)^K\,t^n$ (encoding the dimension of the invariants on the $n$-th symmetric power) of $K$ acting on $V$ coincides with Hilbert–Poincaré series $\sum_{n=0}^{+\infty} \dim(\mathcal{S}^n \mathfrak{a})^W\,t^n$ of the restricted Weyl group $W$ acting on the underlying space $\mathfrak{a}$ of the restricted root system for $G/K$.

This statement is hopefully correct, and certainly very standard (I knew it in the special case $M$ is the underlying space $(G\times G)/G$ of a Lie group, but understood it in the more general case following some useful comments 1 2 3 by Robert Bryant on Invariants for the exceptional complex simple Lie algebra $F_4$). Where can I find it, or as close as possible to it, written in the literature, preferably in a textbook or survey article?

(Note: The reason I have worded it in terms of Hilbert–Poincaré series is that I hope whatever reference comes up will include this as part of a general discussion in the form “the following statement is useful to compute the Hilbert–Poincaré series of certain representations of semisimple groups by reducing them to representations of finite reflection groups; let us now discuss the question of which representations $V$ can be attained that way, and what can be done to compute the Hilbert–Poincaré series of those that cannot”. But maybe I'm asking for too much!)

Answer 1

One reference is in Helgason's 1984 book Groups and Geometric Analysis. The result you want appears there as Corollary 5.12.

The notation he uses is $X=G/K$ is a symmetric space where $G$ is connected and semisimple and $K$ is a maximal compact. As usual, write ${\frak{g}} = {\frak{k}} + {\frak{p}}$ as the Cartan decomposition, let ${\frak{a}}\subset {\frak{p}}$ be a maximal abelian subspace, and let $W$ be the associated Weyl group, i.e., the quotient of the normalizer of ${\frak{a}}$ in $K$ by the centralizer of ${\frak{a}}$ in $K$. Here is what Helgason says:

Corollary 5.12 Every $W$-invariant polynomial $P$ on ${\frak{a}}$ can be uniquely extended to a $K$-invariant polynomial on ${\frak{p}}$.

(Note that the restriction of a $K$-invariant polynomial on ${\frak{p}}$ to ${\frak{a}}$ is clearly $W$-invariant, so we get an isomorphism, and, in particular, an equality of Hilbert-Poincaré series.)

Helgason attributes this result to Chevalley. He calls it "the Chevalley restriction theorem", though he says in the Notes at the end of the Chapter that it is an "unpublished result of Chevalley". He also includes an Exercise (D1) at the end of the Chapter in which he outlines Harish-Chandra's proof of Chevalley's result.