Toeplitz operators, $\mathbb{T}^m$-invariance and quasi-homogeneous symbols

For a partition $\boldsymbol{k} = (k_1, \dots, k_m)$ of $n$ consider the group $\mathrm{U}(\boldsymbol{k}) = \mathrm{U}(k_1) \times \dots \times \mathrm{U}(k_m)$ block diagonally embedded in $\mathrm{U}(n)$ and the center $\mathbb{T}^m$ of $\mathrm{U}(\boldsymbol{k})$. We study the Toeplitz operators with $\mathbb{T}^m$-invariant symbols acting on the weighted Bergman spaces on the unit ball $\mathbb{B}^n$. We introduce the $(\boldsymbol{k},j)$-quasi-radial quasi-homogeneous symbols as those that are invariant under the group $\mathrm{U}(\boldsymbol{k},j,\mathbb{T})$ obtained from $\mathrm{U}(\boldsymbol{k})$ by replacing the factor $\mathrm{U}(k_j)$ with its center $\mathbb{T}$. These symbols are used to build commutative Banach non-$C^*$ algebras generated by Toeplitz operators. These algebras generalize those from the literature and show that they can be built using groups. We describe the action of such Toeplitz operators on monomials through explicit integral formulas involving the symbols. We prove that every Toeplitz operator with $\mathbb{T}^m$-invariant symbol has an associated Toeplitz operator with $\mathrm{U}(\boldsymbol{k})$-invariant symbol in terms of which we can describe some properties.