Let $L=\mathfrak{sl}(2,\mathbb{F})$ with standard basis $(x, y, h)$ and dual basis $(x^{*}, y^{*}, h^{*})$, $H$ a CSA, $W$ the Weyl group and $G=\operatorname{Int}L$.
Let $\mathfrak{P}(L)^{G}$ be the subalgebra of $G$-invariant polynomial functions on $L$, $\mathfrak{P}(H)^{W}$ the subalgebra of $W$-invariant polynomial functions and $\theta: \mathfrak{P}(L)^{G} \to \mathfrak{P}(H)^{W}$ the algebra homomorphism given by restricting $f\in\mathfrak{P}(L)^{G}$ to $H$.
I am trying to prove that $\theta(h^{*2}+x^{*}y^{*})=\lambda^{2}$ for $\lambda=\frac{1}{2}\alpha$ the fundamental dominant weight.
In J. E. Humphreys "Introduction to Lie Algebras and Representation Theory" it is stated, that this is an "easy trace polynomial calculation", but I have no clue where to start.
Thank you very much for helping me.
The polynomial function $f=(h^*)^2+x^* y^*$ takes value $$f(\ell)=h^*(\ell)^2+x^*(\ell) y^*(\ell)$$ on any $\ell \in L$. Evaluating it on $c h$ for a number $c$ therefore gives $$f(ch)=h^*(ch)^2+x^*(h) y^*(h)=c^2+0 \cdot 0=c^2.$$ This is the same as the value of $\lambda^2$ on $ch$ (by definition $\lambda$ is the dual basis element to the positive coroot $h$), so these are the same function on the Cartan subalgebra $\mathbf{C} h$ of $L$.
I also do not understand what is meant by trace polynomial calculation, but it seems to me that the above way of thinking about it is maximally efficient.