**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 by Robert Bryan on [this question](https://mathoverflow.net/questions/96149/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!)