Weyl character formula for the finite dimensional complex semisimple Lie algebras plays a crucial role in the theory of highest weight modules, where for the highest weight module $V(\lambda)$ we definte its charactes as follows. $$ch \, V(\lambda) = \sum_{\mu \in \mathfrak {h}^*} dim \,V_{\mu}(\lambda) \, e^{\mu}.$$

When we go to the theory of highest weight modules for Lie superalgebras we have the same notion of character, but there is an extra super character also which is defined as follows. $$sch \, V(\lambda) = \sum_{\mu \in \mathfrak {h}^*} (dim \,V_{\mu,0}(\lambda)- dim \,V_{\mu,1}(\lambda)) \, e^{\mu}$$

My question is, in the super case I know that $V(\lambda)$ is $\mathbb{Z}_2$ graded and so all its weight spaces also $\mathbb{Z}_2$ graded? and if so why?

What is the motivation for the super character defined above?

Kindly share your views. Thanks a lot.


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