quad V_0(u_i)=u_{i+1},\quad
V_0(v_i)=v_{i-1}.
$$

The vacuum vector in ${\cal F}$ is $\Omega=u_{-1}\wedge v_{-1}\wedge
u_{-2}\wedge v_{-2}\wedge{\ldots} $. We also consider the family of
vectors
$$
\Omega_0=\Omega,\quad \Omega_{2n}=V^{-n}\Omega_0,\quad n\in{\mathbb Z}.
$$

In the space ${\cal F}$ we have a canonical representation of the affine
Lie algebra $\widehat\mathfrak{sl}_2=\mathfrak{sl}_2\otimes{\mathbb C}[t,t^{-1}]\oplus{\mathbb C} c\oplus{\mathbb C} d$, which is given by the following formulas.
Given $x\in\mathfrak{sl}_2$, denote $X(z)=\sum_{n\in{\mathbb Z}}x_nz^{-(n+1)}$. Then
\begin{eqnarray*}
E(z)=\psi(z)\phi^*(z),\qquad
F(z)=\phi(z)\psi^*(z),\\
h_n=a^\psi_{-n}-a^\phi_{-n},\qquad
d=\frac{h_0^2}2+\sum_{n=1}^\infty h_{-n}h_n,\qquad c=1.
\end{eqnarray*}

We have
$$
{\cal F}={\cal H}_0\otimes{\cal K}_0+{\cal H}_1\otimes{\cal K}_1,
$$
where ${\cal H}_0\simeq L_{0,1}$ and ${\cal H}_1\simeq L_{1,1}$ are the
irreducible  level~1 highest weight representations of
$\widehat\mathfrak{sl}_2$ and  ${\cal K}_0$ and ${\cal K}_1$ are the multiplicity spaces.
Observe also that
$$
e_{-(N+1)}\Omega_{-N}=\Omega_{-(N+2)}.
$$

Note that the operators $a_n=\frac1{\sqrt{2}}h_n$
satisfy the CCR~\eqref{CCR}, i.e., form a system of free bosons, or
generate the Heisenberg algebra ${\mathfrak A}_h$.
The vectors $\{\Omega_{2n}\}_{n\in{\mathbb Z}}$ introduced above are exactly singular vectors for
this Heisenberg algebra:
$h_k\Omega_m=0$ for $m<0$, $h_0\Omega_m=m\Omega_m$. The representation of
${\mathfrak A}_h$ in ${\cal H}_0$ breaks into a direct sum of
irreducible representations:
\begin{equation}
{\cal H}_0=\bigoplus_{k\in{\mathbb Z}}{\cal H}_0[2k],
\end{equation}
where ${\cal H}_0[2k]$ is the charge~$2k$ subspace, i.e., the eigenspace of
$h_0$ with eigenvalue $2k$:
$${\cal H}_0[2k]=\{v\in{\cal H}_0:h_0v=2kv\}={\mathbb C}[h_0,h_1,{\ldots} ]\Omega_{2k}.
$$

\subsection{The representation of the Virasoro algebra
associated with the basic representation of $\widehat\mathfrak{sl}_2$}

Given a representation of the affine Lie algebra $\widehat\mathfrak{sl}_2$, we can
use the Sugawara construction to obtain the corresponding
representation of the Virasoro algebra $\operatorname{Vir}$. It can also be
described in the following way.  As noted above, the operators $a_n=\frac1{\sqrt{2}}h_n$
form a system of free bosons. Given
such a system, a representation of $\operatorname{Vir}$ can be constructed as
follows (\cite[Ex.~9.17]{Kac}):
$$
L_n=\frac12\sum_{j\in{\mathbb Z}}a_{-j}a_{j+n},\quad n\ne0;\qquad
L_0=\sum_{j=1}^\infty a_{-j}a_j.
$$
Thus we obtain a representation of $\operatorname{Vir}$ in ${\cal F}$ and, in particular,
in ${\cal H}_0$. In this representation, the algebras generated by the
operators of $\operatorname{Vir}$ and
$\mathfrak{sl}_2\subset\widehat\mathfrak{sl}_2$ are mutual commutants, and we have the decomposition
\begin{equation}\label{virdecomp}
{\cal H}_0=\bigoplus_{k=0}^\infty M_{2k+1}\otimes L(1,k^2),
\end{equation}
where $M_{2k+1}$ is the $(2k+1)$-dimensional irreducible representation of $\mathfrak{sl}_2$
and $L(1,k^2)$ is the irreducible representation of $\operatorname{Vir}$ with
central charge~1 and conformal dimension $k^2$.

The charge $k$ subspace ${\cal H}_0[k]$ contains a series of singular
vectors $\xi_{k,m}$ of $\operatorname{Vir}$  with energy $(k+m)^2$:
$$
L_n\xi_{k,m}=0 \mbox{ for }n=1,2,{\ldots},\qquad L_0\xi_{k,m}=(k+m)^2.
$$

Let us use the so-called
homogeneous vertex operator construction of the basic
representation of $\widehat\mathfrak{sl}_2$ (see \cite[Sec.~14.8]{Kac}). In this realization,
\begin{equation}\label{EF}
E(z)=\Gamma_-(z)\Gamma_+(z)z^{-h_0}V^{-1},\qquad
F(z)=\Gamma_+(z)\Gamma_-(z)z^{h_0}V,
\end{equation}
where
$$
\Gamma_\pm(z)=\exp\left(\mp\sum_{j=1}^\infty \frac{z^{\pm j}}jh_{\pm j}\right)
$$
and the operators $\Gamma_\pm(z)$ satisfy the commutation relation
\begin{equation}\label{Ga}
\Gamma_+(z)\Gamma_-(w)=\Gamma_-(w)\Gamma_+(z)\left(1-\frac zw\right)^2.
\end{equation}

Using the
boson--fermion correspondence (see \cite[Ch.~14]{Kac}), we can
identify ${\cal H}_0$ with the space $\Lambda\otimes{\mathbb C}[q,q^{-1}]$, where $\Lambda$ is
the algebra
of symmetric functions (see \cite{Mac}). In particular, consider
the charge~0 subspace ${\cal H}[0]={\cal H}_0[0]$, which is
identified with $\Lambda$. We can use the following representation of the
Heisenberg algebra generated by $\{h_n\}_{n\in{\mathbb Z}}$:
\begin{equation}\label{power}
h_n\leftrightarrow 2n\frac\partial{\partial p_n},\qquad
h_{-n}=p_n,\qquad n>0,
\end{equation}
where $p_j$ are Newton's power sums. Then
the corresponding Virasoro operators are
\begin{equation}\label{vir0}
\begin{aligned}
L_n&=\sum_{r=n+1}^\infty p_{n-r}\cdot r\frac\partial{\partial p_r}+
\sum_{r=1}^{n-1}r(n-r)\cdot\frac\partial{\partial
p_r}\frac\partial{\partial p_{n-r}},\\
L_{-n}&=\sum_{r=1}^\infty p_{n+r}\cdot r\frac\partial{\partial p_r}+
\frac14\sum_{r=1}^{n-1}p_rp_{n-r},\qquad n>0.
\end{aligned}
\end{equation}
Note that the representation~\eqref{power} of the Heisenberg algebra,
and hence the representation~\eqref{vir0} of the Virasoro algebra, are
not unitary with respect to the standard inner product in $\Lambda$. To
make it unitary, we should consider the inner product in $\Lambda$ defined
by
\begin{equation}
\langle p_\lambda,p_\mu\rangle=\delta_{\lambda\mu}\cdot z_\lambda\cdot 2^{l(\lambda)},
\end{equation}
where $p_\lambda$ are the power sum symmetric functions, $z_\lambda=\prod_i
i^{m_i}m_i!$ for a Young diagram $\lambda$ with $m_i$ parts of length $i$, and
$l(\lambda)$ is the length (number of nonzero rows) of $\lambda$.

Denote the singular vectors of $\operatorname{Vir}$ in ${\cal H}[0]$ by
$\xi_m:=\xi_{0,m}$. According to a result by Segal \cite{Segal}, in
the symmetric function realization~\eqref{