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)