Let $X$ be an abelian variety over an algebraically closed field of charactristic $0$. In this paper, Orlov showed that there is a short exact sequence $$0\to \mathbb Z \oplus X \times \hat X \to\operatorname{Auteq}(Perf X) \to Sp(X\times \hat X) \to 0,$$ where $\operatorname{Auteq}(Perf X)$ denotes the group of triangulated autoequivalences and $Sp(X\times \hat X)$ denotes the group of symplectic (or isometric) automorphisms of $X \times \hat X$. In this question Factorization of symplectic isomorphisms of abelian varieties, the person explained $Sp(X\times \hat X)$ and asked how to fill in an argument right after Proposition 4.12 about factorization of any symplectic automorphism into a composition of two symplectic automorphisms whose factors corresponding to maps from $\hat X$ to $X$ are isogenies, which is used to show the surjectivity in the sequence above and is claimed to be easy. Moreover, this paper addresses the exact issue brought up in the question, mentioning the authors believe the argument is indeed not "easy" unless $X$ is for example simple.

I personally could not fill in this argument either and I am wondering if the argument has been verified elsewhere or known to work out now. I would really appreciate your comments.



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