  We consider the Toeplitz operators on the weighted Bergman spaces over the
unit ball $\mathbb{B}^n$ and their analytic continuation. We proved the
commutativity of the $C^*-$algebras generated by the analytic continuation of
Toeplitz operators with a special class of symbols that satisfy an invariant
property, and we showed that these commutative $C^*-$algebras with symbols
invariant under compact subgroups of $SU(n,1)$ are completely characterized in
terms of restriction to multiplicity free representations. Moreover, we
extended the restriction principal to the analytic continuation case for
suitable maximal abelian subgroups of $SU(n,1)$, we obtained the generalized
Segal-Bargmann transform and we showed that it acts as a convolution operator.
Furthermore, we proved that Toeplitz operators are unitarly equivalent to a
convolution operator and we provided integral formulas for their spectra.
