Character of a finite-dimensional representation of a semi-simple Lie algebra

The function that assigns to every weight of the representation the dimension of the corresponding weight subspace. If $\mathfrak h$ is a Cartan subalgebra of a semi-simple Lie algebra $\mathfrak g$ over an algebraically closed field $k$ of characteristic $0$, $\phi : \mathfrak g \rightarrow \mathfrak g \mathfrak l (V)$ is a linear representation and $V _ \lambda$ is the weight subspace corresponding to $\lambda \in \mathfrak h ^ {*}$, then the character of the representation $\phi$ (or of the $\mathfrak g$-module $V$) can be written in the form

$$\mathop{\rm ch} V = \ \sum _ {\lambda \in \mathfrak h ^ {*} } ( \mathop{\rm dim} V _ \lambda ) e ^ \lambda$$

and can be regarded as an element of the group ring $\mathbf Z [ \mathfrak h ^ {*} ]$. If $k = \mathbf C$ and $\phi = d \Phi$, where $\Phi : G \rightarrow \mathop{\rm GL} (V)$ is an analytic linear representation of a Lie group $G$ with Lie algebra $\mathfrak g$, then $e ^ \lambda$ can be regarded as the function on $\mathfrak h$ suggested by the notation and $\mathop{\rm ch} \phi$ coincides with the function $x \mapsto \chi _ \Phi ( e ^ {x} )$ ($x \in \mathfrak h$), where $\chi _ \Phi$ is the character of the representation $\Phi$. Characters of a representation of a Lie algebra have the following properties:

$$\mathop{\rm ch} (V _ {1} \oplus V _ {2} ) = \ \mathop{\rm ch} V _ {1} + \mathop{\rm ch} V _ {2} ,$$

$$\mathop{\rm ch} (V _ {1} \otimes V _ {2} ) = \mathop{\rm ch} V _ {1} \cdot \mathop{\rm ch} V _ {2} .$$

References

[1]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[2]	J. Dixmier, "Algèbres enveloppantes" , Gauthier-Villars (1974)