\tau + N + \tfrac{1}{2}, q + \tfrac{1}{2}\rangle = [N + (n+a+b)k + 1]^{-1/2} V_+ |\tau, \tau + N, q
         \rangle, \\
  & |-, \tau + N - \tfrac{1}{2}, q + \tfrac{1}{2}\rangle = (N + nk)^{-1/2} V_- |\tau, \tau + N, q\rangle, \\
  & |\pm, \tau + N, q + 1\rangle = [n (n+a+b)k^2]^{-1/2} V_+ V_- |\tau, \tau + N, q\rangle.
\end{split}  \label{eq:normalized-fermionic}$$ It can also be shown that the one-fermion states with the same eigenvalue of $K_0$ are not orthogonal and that their overlap is given by $$\langle +, \tau + N - \tfrac{1}{2}, q + \tfrac{1}{2} | -, \tau + N - \tfrac{1}{2}, q + \tfrac{1}{2}\rangle =
  \left(\frac{N [N + (2n+a+b)k]}{[N + (n+a+b)k] (N+nk)}\right)^{1/2}$$ for $N=1$, 2 ….

The results obtained so far for one- and two-fermion states correspond to generic values of $N$ and $n$. To construct from them basis states for ${\rm sp}(2, {{\mathbb{R}}})$ irreps we have to distinguish between vanishing and non-vanishing values of $n$.

For $n=0$, as a result of (\[eq:V-appl\]) and of the properties $Z^{(a+b)}_{-1}(z) = \Phi^{(a+1,b+1)}_{-1}(\varphi) = 0$, it turns out that the ground state $|\tau, \tau, q\rangle = \Psi_{0,0} |0\rangle$ of ${\cal H}^s$ is annihilated not only by $K_-$ and $W_-$, 