Journal of Lie Theory, Vol. 10, No. 2, pp. 323-330 (2000)

On the Chevalley restriction theorem

E. A. Tevelev

Moscow State University
Chair of Algebra
119899, Vorob'evy Gory, Moscow, Russia

Abstract: Let $\g$ be a complex semisimple Lie algebra with adjoint group $G$. Suppose that $\sigma$ is an involutive automorphism of $\g$. Then $\sigma$ induces an involution of $G$ also denoted by $\sigma$. Let $K=G^\sigma$ be the subgroup of $\sigma$-fixed points. Consider the direct decomposition $\g=\k\oplus\p$ of $\g$ into the eigenspaces for $\sigma$, so that $\k=\{x\in\g | \sigma(x)=x\}$, $\p=\{x\in\g | \sigma(x)=-x\}$. Then $\p$ is a $K$-module. Denote by $\a\subset\p$ a maximal abelian ad-diagonalizable subalgebra. Consider the "baby Weyl group" $W_0=N_K(\a)/Z_K(\a)$. The aim of this paper is to prove that the restriction map $\psi: \C[\p\times\p]^K\to\C[\a\times\a]^{W_0}$ is surjective.

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