TITLE: Applications of Chevalley Restriction Theorem
QUESTION [10 upvotes]: Let $G$ be a simple linear algebraic group (over $\mathbb{C}$, say) and $\mathfrak{g}$ be its Lie algebra, $\mathfrak{t}\subset \mathfrak{g}$ the Lie algebra of a maximal torus in $G$ and $W$ the corresponding Weyl group. The Chevalley Restriction Theorem (CRT) states that there is an isomorphism
$$ \mathbb{C}[\mathfrak{g}]^{G}\simeq \mathbb{C}[\mathfrak{t}]^{W}$$
obtained by restriction. This is the Lie-theoretic statement: $G$-invariant functions on $\mathfrak{g}$ are trace functions (ie, sums of 'eigenvalues'). Admittedly, this is a result that I feel I understand quite well - and that intuitively makes sense - but I don't know too many applications of it.
Question: what are some (nontrivial) applications of the above isomorphism? (ie, what cool things can be proved using the CRT?)
For example, CRT can be used to show that the defining ideal of the nilpotent cone $N\subset \mathfrak{g}$ is $\mathbb{C}[g]^{G}_{+}$, the ideal generated by the $G$-invariant functions without constant term (this is in Chriss & Ginzburg, Ch. 3; although there are proofs of this fact bypassing CRT), and this can be used to show that the algebra defining the intersection $N\times_{\mathfrak{g}} \mathfrak{t}$ is the coinvariant algebra $\mathbb{C}[\mathfrak{t}]/\mathbb{C}[\mathfrak{t}]^{W}_{+}$. 
EDIT 1: As mentioned in Konstantin Ardakov's answer, the CRT can be applied to prove the Harish-Chandra description of the centre of the universal enveloping algebra. I was preferably looking for straight applications of the CRT as opposed to generalisations, but I welcome all comments/suggestions that people may have.

REPLY [7 votes]: Let $G$ be a split reductive group over a finite field $k$ (of sufficiently large characteristic), $X$ a smooth geometrically connected projective curve over $k$ and $\mathcal{M}$ the algebraic stack that classifies Hitchin pairs on $X$ (see reference below, but these are pairs $(E,\varphi)$ where $E$ is a $G$-bundle on $X$ and $\varphi$ is a section of a certain bundle derived from $E$). 
The Chevalley restriction map for $G$ suitably twisted over $X$ is essentially the the Hitchin fibration $f:\mathcal{M}\to A$ where $A$ is an affine space; for nice enough $a\in A$ the fibers $f^{-1}(a)$ can be written as a nice sum of orbital integrals related to the ones appearing in various trace formulae for groups over number fields, and this is an essential tool in Ngô's proof of the fundamental lemma.
Here, twisted over $X$ means twisted by a torsor that gives the data for a group scheme on $X$ locally isomorphic to $X\times_k G$ in the étale topology.
See for instance Ngô's paper "Fibration de Hitchin et Endoscopie".