Hi:
Given a finite dimensional Lie superalgebra $G$ over the field of complex numbers with decomposition $G_{-1}+G_{0}+G_{1}$, where $G_{0}$ is the even part and $G_{-1}+G_{1}$ is the odd part. Suppose $F$ is the category of all finite dimensional weight $G_{0}$-modules. Is it a general fact that all irreducible (or indecomposable) highest weight modules in $F$ are projective ? And, I'm not sure if the answer depends on the type of $G$.
Thanks!!
Then the answer is yes, finite dimensional irreducible $G_0$-modules are projective in the category of finite dimensional modules.
Note that $G_0=G_0'\oplus Z$, where $G_0'$ is semisimple and $Z$ is the center. Let $V$ be a finite dimensional $G_0$-module. Then, by Weyl's theorem, $V$ is completely irreducible as a $G_0'$-module. The elements of $Z$ act as scalars on each irreducible summand of $V$, by Schur's lemma. This means that the decomposition of $V$ into irreducible $G_0'$-modules is a decomposition as $G_0$-modules. It now follows that the category of finite dimensional $G_0$-modules is semi-simple, and irreducible=projective.
EDIT: This answer is based on the assumption that $G$ is a simple Lie superalgebra, so $G_0$ is reductive. At one time, there was a comment/answer indicating that this was the case of interest.
