E_{N,n} - E_{0,0} = 4\omega (N + nk).$$ Such extended wavefunctions turn out to be also eigenstates of the ${\rm osp}(2/2, {{\mathbb{R}}})$ weight generators $K_0$ and $Y$ corresponding to the eigenvalues $\tau + N$ and $q$, where $$\tau = \left(n + \frac{a+b}{2}\right)k + \frac{1}{2}, \qquad q = - \frac{1}{2}[(a+b)k + 1].$$

All the states $\Psi_{N,n} |0\rangle$ with a definite value of $n$ (hence of $\tau$) and $N=0$, 1, 2, … belong to a ${\rm sp}(2, {{\mathbb{R}}})$ lowest-weight state (LWS) irrep characterized by $\tau$ and will be denoted by $$|\tau, \tau + N, q\rangle = \Psi_{N,n} |0\rangle.  \label{eq:zero-fermion}$$ They indeed satisfy the relations $$\begin{split}
  & K_0 |\tau, \tau + N, q\rangle = (\tau + N) |\tau, \tau + N, q\rangle, \\
  & K_+ |\tau, \tau + N, q\rangle = [(N + 1)(2\tau + N)]^{1/2} |\tau, \tau + N + 1, q\rangle, \\
  & K_- |\tau, \tau + N, q\rangle = [N(2\tau + N - 1)]^{1/2} |\tau, \tau + N - 1, q\rangle,
\end{split}$$ which can be easily checked by rewriting $K_{\pm}$ as $$K_{\pm} = - \frac{1}{4\omega} H_k + \frac{1}{2} z \mp \left(z \partial_z + \frac{1}{2}\right) - \Gamma$$ and using some well-known properties of Laguerre polynomials [@gradshteyn].

The odd generators $W_{\pm}$ annihilate 