Invariant form on Lie algebra

Question

Does anyone have a reference for the following fact?

Let $G \subset GL(n)$ be a compact Lie group. Then the form $$f(A)=-Tr(A^2)$$ defined for $A \in T_e G \subset \mathfrak{gl}(n)$ is positive definite. I'm looking for a direct proof.

Answer 1

I'm not sure whether this is direct enough for your liking, but it might be helpful! The standard proof uses the "unitarian trick" of Weyl. One first introduces a bi-invariant measure on the Lie group $G$, possible since the group is compact.

Now if $\mathcal{D}$ is a representation of $G$ on $V$ one obtains a positive definite Hermitian inner product on $V$ by integration over the group, and this is invariant under $\mathcal{D}$. With respect to this inner product the representation is unitary, so it's Lie algebra is antihermitian.

Therefore the eigenvalues of any Lie algebra element are pure imaginary. Thus the trace of the square of a Lie algebra element is negative definite, proving the result.

For a more detailed argument, and a proof of the converse see p154 of Sattinger & Weaver "Lie Groups and Algebras with Applications to Physics, Geometry and Mechanics".