Killing form in relation to character of adjoint representation

Question

The Lie theory 'killing form' feels very similar to just being the character of the adjoint representation, of course taking two inputs rather than one. How do these relate?

Answer 1

First note that, while group characters are defined as $\chi_\rho(s) = \mathrm{Tr}(\rho_s)$ for some representation $\rho$ and group element $s$, Lie algebra characters are defined slightly differently. For some representaiton $\theta: \mathfrak g \to \mathfrak{gl}(V)$ of some Lie algebra $\mathfrak g$ (can be assumed to be semisimple for simplicity here), a character $\chi_\theta$ is defined as a function over the Cartan subalgebra $\mathfrak h$ of $\mathfrak g$ in a way that for any element $h \in \mathfrak h$ you get $\chi_\theta(h) = \mathrm{Tr} e^{\theta(h)}$. So essentially they end up being the trace of the metrix representative of the Lie group element $\mathrm{exp}(h) \in \mathrm{GL}(V)$. (c.f. [Wiki]).

So, the similarity that you mentioned is less prominent, since the algebraic character evaluates to a trace of a group element and the Killing form uses trace of the matrix product of the matrix representatives of two algebra elemetns. Also note that for any two element $a, b \in \mathfrak g$ the Killing form evaluates $\mathrm{Tr}(\theta(a) \circ \theta(b))$ where $\theta$ is the adjoint representation, and $\rho(a) \circ \rho(b)$ is not a representative of any algebra element, so the trace is over something that doesn't belong to the algebra. The sole purpose of the Killing form is to give you an (extremely useful) inner product for the Lie algebra that doesn't require any extrinsic representation theoretic data (the data defining the adjoint representation is equivalent to the data defining the Lie algebra, you don't need anything extra, so use of the adjoint representation is acceptable here). The prupose of the characters, on the other hand, is to encode data that characterizes various representations. For example, the data of the characters of semi simple complex Lie algebras are equivalent to the data defining their finite dimensional representations (c.f. [Ser]:Chapter VII, Proposition 5). Thus, conceptually the Killing form and the characters are essentially different objects.