Boundary representations on co-invariant subspaces of Bergman space

Let $M$ be an invariant subspace of the Bergman space $L_a^2(\mathbb{D})$ and $S_M$ be the compression of the coordinate multiplication operator $M_z$ to the co-invariant subspace $L_a^2(\mathbb{D})\ominus M$. The present paper determines when the identity representation of $C^*(S_M)$ is a boundary representation for the Banach subalgebra $\mathcal{B}(S_M)$. The paper also considers boundary representations on the co-invariant subspaces of $L_a^2(\mathbb{B}_n)$.