  On the unit ball B^n we consider the weighted Bergman spaces H_\lambda and
their Toeplitz operators with bounded symbols. It is known from our previous
work that if a closed subgroup H of \widetilde{\SU(n,1)} has a
multiplicity-free restriction for the holomorphic discrete series of
$\widetilde{\SU(n,1)}$, then the family of Toeplitz operators with H-invariant
symbols pairwise commute. In this work we consider the case of maximal abelian
subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise
commutativity of the corresponding Toeplitz operators. To achieve this we
explicitly develop the restriction principle for each (conjugacy class of)
maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform.
In particular, we obtain a multiplicity one result for the restriction of the
holomorphic discrete series to all maximal abelian subgroups. We also observe
that the Segal-Bargman transform is (up to a unitary transformation) a
convolution operator against a function that we write down explicitly for each
case. This can be used to obtain the explicit simultaneous diagonalization of
Toeplitz operators whose symbols are invariant by one of these maximal abelian
subgroups.
