# Differential of a nilpotent or semisimple element

Let $G$ be an algebraic connected subgroup of $GL_n(\mathbb{C})$ and let $\chi : G \to \mathbb{C}^*$ a character. Consider $d_e\chi : \mathfrak{g} \to \mathbb{C}$ the differential of $\chi$ at the identity of $G$. Is it true that $d_e\chi(\nu)=0$ if $\nu$ is a nilpotent element?If not, is it true under some assumptions? Can we say something if $\nu$ is semisimple?

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The question should be formulated more precisely. The assumption seems to be that $G$ is an algebraic group and that $\chi$ is a morphism of algebraic groups (?) If so, assuming as we may that $\chi$ is nontrivial, $\chi$ induces an isomorphism of a 1-dimensional quotient of $G$ (a torus) onto the image . The corresponding 1-dimensional quotient of the Lie algebra is then isomorphic to the Lie algebra of the 1-torus and thus consists of semisimple elements. In particular, all nilpotent elements of $\mathfrak{g}$ must lie in the kernel of $d_e\chi$. (This uses the algebraic theory, with Chevalley's version of Jordan decomposition in both the group and its Lie algebra along with good behavior of quotients. There are some similar ideas in prime characteristic, but with added complications.)
$\chi$ is a character attached to a semi-invariant of the action of $G$ on $\mathbb{C}^n$. thank you. –  Michele Torielli Feb 18 '11 at 14:38