Adding complex fermions to the Grassmannian-like coset model

In the N = 2 supersymmetric coset model, $\frac{SU(N+M{)}_k\times SO(2NM{)}_1}{SU(N{)}_{k+M}\times U(1{)}_{NM(N+M)(k+N+M)}}$, we construct the SU(M) nonsinglet N = 2 multiplet of spins (1, $\frac{3}{2}$, $\frac{3}{2}$, 2) in terms of coset fields. The next SU(M) singlet and nonsinglet N = 2 multiplets of spins (2, $\frac{5}{2}$, $\frac{5}{2}$, 3) are determined by applying the N = 2 supersymmetry currents of spin $\frac{3}{2}$ to the bosonic singlet and nonsinglet currents of spin 3 in the bosonic coset model. We also obtain the operator product expansions (OPEs) between the currents of the N = 2 superconformal algebra and above three kinds of N = 2 multiplets. These currents in two dimensions play the role of the asymptotic symmetry, as the generators of N = 2 "rectangular W-algebra", of the M $\times$ M matrix generalization of N = 2 AdS${}_3$ higher spin theory in the bulk. The structure constants in the right hand sides of these OPEs are dependent on the three parameters k, N and M explicitly. Moreover, the OPEs between SU(M) nonsinglet N = 2 multiplet of spins (1, $\frac{3}{2}$, $\frac{3}{2}$, 2) and itself are analyzed in detail. The complete OPE between the lowest component of the SU(M) singlet N = 2 multiplet of spins (2, $\frac{5}{2}$, $\frac{5}{2}$, 3) and itself is described. In particular, when M=2, it is known that the above N = 2 supersymmetric coset model provides the realization of the extension of the large N = 4 nonlinear superconformal algebra. We determine the currents of the large N = 4 nonlinear superconformal algebra and the higher spin-$\frac{3}{2}$, 2 currents of the lowest N = 4 multiplet for generic k and N in terms of the coset fields. For the remaining higher spin-$\frac{5}{2}$, 3 currents of the lowest N = 4 multiplet, some of the results are given.