# Applications of Chevalley Restriction Theorem

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.

• @George: Maybe you should add a note to the question along the lines of your comment on Ardakov's answer? Anyway I guess you are not expecting a unique correct answer, but it seems you are also getting some pointers to generalizations rather than applications of the CRT (which occur in a number of theories). – Jim Humphreys Jun 1 '13 at 14:46
• @Jim: Thanks for the suggestion, I have edited the question to emphasise the preference for applications of the CRT. – George Melvin Jun 1 '13 at 20:11

How about the Harish-Chandra isomorphism, which computes the centre of the universal enveloping algebra of $\mathfrak{g}$?

• Of course! I knew of this (the first place I saw the use of the CRT) but forgot to include it in my known examples in the question. Thanks for the gentle reminder though :) – George Melvin May 31 '13 at 22:46

The ideas of the Chevalley restriction theorem have been generalized by Solomon to polynomial differential forms, by Palais and Terng to smooth functions, and in the papers 1, 2 (see also references therein), to prove the following result:

• For a proper isometric action of a Lie group $G$ on a smooth Riemannian manifold $M$ admitting a section $\Sigma$ the restriction of differential forms induces an isomorphism $$\Omega^p_{\text{hor}}(M)^G \cong \Omega^p(\Sigma)^{W(\Sigma)}$$ between the space of horizontal $G$-invariant differential forms on $M$ and the space of all differential forms on $\Sigma$ which are invariant under the action of the generalized Weyl group $W(\Sigma)$ of the section $\Sigma$.

This has been carried over to the algebraic geometry setting in the paper:

• MR1451789 (98k:14067)
Brion, Michel(F-GREN-F) Differential forms on quotients by reductive group actions. (English summary) Proc. Amer. Math. Soc. 126 (1998), no. 9, 2535–2539.
• @Peter Michor: Thanks for your comment. These are new results to me; I had seen some generalisations of the CRT due to Broer in the 'vector-valued functions' case (using 'small' representations). Also, thanks for the reference to Brion's article. – George Melvin Jun 1 '13 at 20:21

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".

• @Jason: Thanks for your comment; this looks particularly interesting. – George Melvin Jun 3 '13 at 19:26

I think the most important and direct generalization of the Chevalley Restriction Theorem is the Luna–Richardson Restriction Theorem. Suppose that you have a representation $V$ of a reductive algebraic group $G$, and $x \in V$ is a point mapping to a principal point of $V/G$, which means that for all points in some open subset of $V$ containing $x$, the isotropy (= stabilizer) subgroups are conjugate; this would be guaranteed if $G\cdot x$ is closed and has maximal dimension among all orbits. Let $H$ be the isotropy subgroup of $x$. The Luna–Richardson Restriction Theorem tells you that the restriction map gives an isomorphism between the two rings of invariants $F[V]^G$ and $F[V^H]^{N_G(H)/H}$.

This can be extremely useful as it is often hard to compute the first ring directly but easy to compute the second. This is what made the Chevalley Restriction Theorem important. The $G$-invariant functions on the Lie algebra are hard to find but by the theorem, one reduces to working with a representation of a finite group—a much easier problem (for this case).

Here is the reference for the Luna–Richardson Restriction Theorem:

MR0544240 (80k:14049)

Luna, D.; Richardson, R. W.

Duke Math. J. 46 (1979), no. 3, 487–496. 14L30 (15A72)

• @David: Thanks for your comment and reference. – George Melvin Jun 3 '13 at 19:24

Note that both algebras are polynomial rings, i.e. free commutative algebras. Thus knowing that one of the two sides of the isomorphism is a polynomial ring implies that the other is, too. For $G$ classical, this can be explicitly verified for both the left hand side (using the characteristic polynomial) and the right hand side (using the main theorem on elementary symmetric functions), but the proofs are quite different.

As an application of the Chevalley Restriction Theorem, we get a structure theorem for adjoint invariants in arbitrary semisimple $G$ setting.

The $Ad(G)$-invariant polynomial functions on $\mathfrak{g}$ form a polynomial ring in $\operatorname{rank}(G)$ variables.

This is true because the $W$ is a reflection group, and by the Chevalley-Shephard-Todd theorem, the right hand side $\mathbb{C}[\mathfrak{t}]^W$ is a polynomial ring.

• Another application, for which I don't have time now, is Tanisaki's description (in the general linear case) of the scheme-theoretic intersection of the closure of a nilpotent orbit with the Cartan subalgebra . – Victor Protsak Jun 1 '13 at 1:00
• @Victor Protsak: Thanks for your comments. I was aware that $\mathfrak{g}//G$ was an affine space alongs the lines that you mention, but wonder of any interesting consequences of this result? – George Melvin Jun 1 '13 at 20:16
• "Interesting" is a subjective term. For example, actions with the polynomial ring of invariants (FA) are in many ways the nicest actions; for the case when a connected simple algebraic group $G$ acts linearly on $V$, they have all been classified. If, in addition to FA, the action is equidimensional, i.e. the scheme-theoretic fibers of the factorization map $V\to V/G$ all have the same dimension (ED), then the algebra of regular functions $k[V]$ is a free module over the ring of the invariants $k[V]^G$ (FM). This is one way to prove Kostant's theorem that the adjoint action satisfies (FM). – Victor Protsak Jun 3 '13 at 21:02
• @VictorProtsak, do you happen to have a reference for the result of Tanisaki? Is it Tanisaki - Defining ideals of the closures of conjugacy classes and representations of the Weyl groups? – LSpice May 21 at 21:28