Chevalley restriction theorem

In the mathematical theory of Lie groups, the Chevalley restriction theorem describes functions on a Lie algebra which are invariant under the action of a Lie group in terms of functions on a Cartan subalgebra.

Chevalley's theorem requires the following notation:

	assumption	example
G	complex connected semisimple Lie group	SLn, the special linear group
${\displaystyle {\mathfrak {g}}}$	the Lie algebra of G	${\displaystyle {\mathfrak {sl}}_{n}}$, the Lie algebra of matrices with trace zero
${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}}$	the polynomial functions on ${\displaystyle {\mathfrak {g}}}$ which are invariant under the adjoint G-action
${\displaystyle {\mathfrak {h}}}$	a Cartan subalgebra of ${\displaystyle {\mathfrak {g}}}$	the subalgebra of diagonal matrices with trace 0
W	the Weyl group of G	the symmetric group Sn
${\displaystyle \mathbb {C} [{\mathfrak {h}}]^{W}}$	the polynomial functions on ${\displaystyle {\mathfrak {h}}}$ which are invariant under the natural action of W	polynomials f on the space ${\displaystyle \{x_{1},\dots ,x_{n},\sum x_{i}=0\}}$ which are invariant under all permutations of the xi

Chevalley's theorem asserts that the restriction of polynomial functions induces an isomorphism

${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}\cong \mathbb {C} [{\mathfrak {h}}]^{W}}$.

Humphreys (1980) gives a proof using properties of representations of highest weight. Chriss & Ginzburg (2010) give a proof of Chevalley's theorem exploiting the geometric properties of the map ${\displaystyle {\widetilde {\mathfrak {g}}}:=G\times _{B}{\mathfrak {b}}\to {\mathfrak {g}}}$.

This theorem in fact more generally holds for any reductive Lie algebra, and can also be proved directly in the case that ${\displaystyle G=\mathrm {GL} _{n}}$. In this case, the Cartan subalgebra ${\displaystyle {\mathfrak {t}}}$ can be identified with diagonal matrices, and the Weyl group can be identified with the symmetric group acting on diagonal matrices by permutations. Therefore, ${\displaystyle \mathbb {C} [{\mathfrak {t}}]\cong \mathbb {C} [x_{1},...,x_{n}]}$ and ${\displaystyle \mathbb {C} [{\mathfrak {t}}]^{W}}$ can be identified with the subring of ${\displaystyle \mathbb {C} [{\mathfrak {t}}]}$ generated by ${\displaystyle e_{1},...,e_{n}}$ where each ${\displaystyle e_{i}}$ is an elementary symmetric polynomial.

Identify ${\displaystyle {\mathfrak {g}}}$ with the set of ${\displaystyle n\times n}$ matrices. For injectivity, if we are given a ${\displaystyle G}$-invariant polynomial on ${\displaystyle {\mathfrak {g}}}$ which vanishes on the set of diagonal matrices, by ${\displaystyle G}$-invariance, this polynomial vanishes on any matrix which is conjugate to a diagonal matrix. Since these elements are Zariski dense in ${\displaystyle {\mathfrak {g}}}$, this polynomial vanishes on a Zariski dense open subset of ${\displaystyle {\mathfrak {g}}}$ and thus must itself be the zero polynomial.

For surjectivity, it of course suffices to show that each ${\displaystyle e_{i}}$ lies in the image of this map. Using Newton's identities and an inductive argument, it suffices to prove that the function ${\displaystyle x_{1}^{i}+...+x_{n}^{i}}$ lies in the image of this map for every ${\displaystyle i\in \{1,...,n\}}$. However, in this case, one can explicitly construct an invariant function which restricts to ${\displaystyle x_{1}^{i}+...+x_{n}^{i}}$ on diagonal matrices: it is given by the function ${\displaystyle A\mapsto \mathrm {trace} (A^{i})}$.

• Chriss, Neil; Ginzburg, Victor (2010). Representation theory and complex geometry. Birkhäuser. doi:10.1007/978-0-8176-4938-8. ISBN 978-0-8176-4937-1. S2CID 14890248. Zbl 1185.22001.
• Humphreys, James E. (1980). Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics. Vol. 9. Springer. Zbl 0447.17002.