Sufficient conditions for unitarity of a representation of a Lie Superalgebra

Question

Suppose we have a Lie superalgebra with triangular decomposition: \begin{equation} \mathfrak{g} = \mathfrak{g}^{+} \oplus \mathfrak{g}^{0} \oplus \mathfrak{g}^{-} \end{equation}

I've seen it stated in a few places that a sufficient condition for unitarity of a lowest weight representation of a lie superalgebra formed from quotienting out null states is just checking that all states in \begin{equation} U(\mathfrak{g}_1^{+})v_{\lambda} \end{equation} have positive norm, where $v_{\lambda}$ is the lowest weight state, and $\mathfrak{g}_1^{+}$ are the fermionic raising operators. I'm using the candidate inner product given by setting $<v_{\lambda}, v_{\lambda}> =1$ and computing all other norms in the inner product by using the (anti)commutation relations of the algebra.

An example paper for reference: https://arxiv.org/pdf/hep-th/0201076.pdf (section 2.6).

I can't see why this is true. All I can see is that this condition would imply that there is a basis of the Verma module, each of which with positive norm.

Answer 1

If i have correctly understood your question, i think that the answer can be found at

M. D. Gould, R. B. Zhang, Classification of all star and grade star irreps of gl(n|1), J. of Math. Phys., 31, 1524 (1990).

See particularly Lemma 2, p. 1526 and the following discussion (esp. at the end of p. 1527).
(I think the authors there use the term "star representations" as identical to "unitary representations").