On the adjoint quotient of Chevalley groups over arbitrary base schemes

Abstract

AbstractFor a split semisimple Chevalley group scheme G with Lie algebra $\mathfrak{g}$ over an arbitrary base scheme S, we consider the quotient of $\mathfrak{g} by the adjoint action of G. We study in detail the structure of $\mathfrak{g} over S. Given a maximal torus T with Lie algebra $\mathfrak{t}$ and associated Weyl group W, we show that the Chevalley morphism π : $\mathfrak{t}$/W → $\mathfrak{g}/G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient $\mathfrak{g} → \mathfrak{g}//G commutes, or does not commute, with base change on S.