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Let $G$ be a linear algebraic group and $\phi$ a finite dimensional regular representation of $G$ into $GL(V)$

I would like to know about bilinear forms on $V$ and when they are $G$-invariant. Specifically, I want to show that a bilinear form $Z$ on $V$ is $G$-invariant if and only if

$Z(d\phi(X)(v),w)=-Z(v,d\phi(X)(w)) \\ \forall v, w \in V, and \forall X \in Lie(G)$

By $Lie(G)$ I mean the Lie algebra.

My definition of a $G$-invariant bilinear form is that $Z(\phi(g)(v),\phi(g)(w))=Z(v,w)$ for all $v,w$ in $V$ and for all $g \in G$. In the direction assuming that $Z$ is $G$-invariant, I don't see how to involve the Lie algebra. I also tried to think about where the minus sign could come up, to maybe find some idea in that way, but I could not solve it.

Assuming the identity, I am again faced with Lie algebras, where I need information about the representations of the group.

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1 Answer 1

Just take the derivative of $Z(\phi(g)(v),\phi(g)(w) )=Z(v,w)$ and we get $\frac{d}{dt}Z(\phi(exp(tX))(v),\phi(exp(tX))(w))=0=Z(d\phi(X)(v),w)+Z(v, d\phi(X)(w)) $

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