It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant polynomials $k[P]^W$ is a polynomial algebra. Moreover, the generators of this polynomial algebra are associated to the fundamental weights $\omega_1,\dots,\omega_n$ of the root system.
For example, if $\Phi$ is the root system of a group $G$ with maximal torus then this polynomial algebra is $k[T]^W$, which is generated by the traces of the fundamental representations $\mathrm{tr}(\rho_1),\dots,\mathrm{tr}(\rho_n)$, which are eigenfunctions of $T$ with weight $\omega_i$.
My question is to what extent a result of this kind can be obtained when $\Phi$ is non-reduced.
Some discussion on this direction appears in Richardson's article [2] Sections 13, 14 and 15. There, Richardson studies the algebra $k[A]^{W_0}$, where $A$ is a maximal $\theta$-split torus for some involution $\theta$ of a reductive group $G$ and $W_0$ is the corresponding "little" Weyl group. However, there is not a discussion in that paper about the generators of this algebra when it is a polynomial algebra. In the lines of the above, I would expect the generators to be eigenfunctions $f_i\in k[G/G^\theta]$ with weights $\eta_i$ the fundamental weights as constructed, for example, in Vust [3].
References
[1] Lie Groups and Lie Algebras. N. Bourbaki.
[2] Orbits, invariants, and representations associated to involutions of reductive groups. R. W. Richardson.
[3] Plongements d'espaces symétriques algébriques: une classification. T. Vust.
