Is the $\ell$-Diffie Hellman Inversion easy when g is known? From here they define the $\ell$-Diffie Hellman inversion problem as:


  Given $g^{a},g^{a^2}\ldots,g^{a^{\ell}} \in G$, compute $g^{a^{-1}}$


Would this problem become easy if the generator $g$ is also known?
No.

The stated $l$-DHI problem is believed to remain hard when $g$ is known.

Actually, in the quoted page, it is assumed that $g$ is known throughout. This is obvious in particular in the sections on DLP and CDH.