Torus actions on symplectic orbi-spaces

Which $2n$-dimensional orbi-spaces have effective symplectic $k$- torus actions? As shown by Lerman and Tolman (1997) and Watson (1997), this question reduces to that of characterizing the finite subgroups of centralizers of tori in the real symplectic group $Sp(2n, \mathbb{R})$. We resolve this question, and generalize our method to a calculation of the centralizers of all tori in $Sp(2n,\mathbb{R})$.