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.

generalizationsrather than applications of the CRT (which occur in a number of theories). $\endgroup$ – Jim Humphreys Jun 1 '13 at 14:46