Virasoro Verma modules and contravariant forms

The three subspaces ${\mathfrak{n}}_{-}$, $\mathfrak{h}$, and ${\mathfrak{n}}_{+}$ are Lie subalgebras by [F1], and their sum is direct by the Virasoro basis. Since $[L_0,L_n]=-n{L}_n$ and $C$ is central, $[\mathfrak{h},{\mathfrak{n}}_{+}]\subseteq {\mathfrak{n}}_{+}$; also $[{\mathfrak{n}}_{+},{\mathfrak{n}}_{+}]\subseteq {\mathfrak{n}}_{+}$ and $[\mathfrak{h},\mathfrak{h}]=0$. Thus $\mathfrak{b}$ is a Lie subalgebra and the linear functional sending $L_0,C,L_n$ ($n>0$) to $h,c,0$ kills $[\mathfrak{b},\mathfrak{b}]$. By the quotient relations [F2] it extends to a character of $U(\mathfrak{b})$, defining ${\mathbb{C}}_{c,h}$; its restriction to the commuting $L_0,C$ defines ${\chi }_{c,h}$. The assignment $\sigma (L_n)=L_{-n}$, $\sigma (C)=C$ satisfies $\sigma ([L_m,L_n])=-[L_{-m},L_{-n}]$: the noncentral terms agree, and when $m+n=0$ the central coefficient on the right is $(m^3-m)\mathrm{/}12$. Reversing tensor words therefore preserves the ideal in [F2], descends to $U(\mathrm{V}\mathrm{i}\mathrm{r})$, and squares to the identity. Order a countable basis in these three consecutive blocks. PBW [F3] then makes multiplication a vector-space isomorphism $U({\mathfrak{n}}_{-})\otimes U(\mathfrak{h})\otimes U({\mathfrak{n}}_{+})\to U(\mathrm{V}\mathrm{i}\mathrm{r})$. In particular, as a right $U(\mathfrak{b})$-module, $U(\mathrm{V}\mathrm{i}\mathrm{r})$ is free with basis the ordered ${\mathfrak{n}}_{-}$-monomials. Tensoring over $U(\mathfrak{b})$ with ${\mathbb{C}}_{c,h}$ therefore identifies $M(c,h)$ with $U({\mathfrak{n}}_{-})v_h$; the top vector is nonzero.

Let $\pi :U(\mathrm{V}\mathrm{i}\mathrm{r})\to U(\mathfrak{h})$ be the PBW projection which retains the term with no ${\mathfrak{n}}_{-}$ or ${\mathfrak{n}}_{+}$ factor, and let ${\chi }_{c,h}:U(\mathfrak{h})\to \mathbb{C}$ be the algebra homomorphism $L_0\mapsto h$, $C\mapsto c$. For $x,y\in U({\mathfrak{n}}_{-})$, define $⟨x{v}_h,y{v}_h⟩={\chi }_{c,h}(\pi (\sigma (x)y))$. The free-basis identification in step 1.1 makes this a well-defined bilinear form on the induced module, and $⟨v_h,v_h⟩=1$. This formula also determines every pairing from the top normalization by repeatedly moving negative modes to the second argument; hence the normalized contravariant form, if contravariant, is unique.

For an ordered PBW monomial $x_{-}a{x}_{+}$, with factors in the three blocks, its image under $\sigma$ is $\sigma (x_{+})\sigma (a)\sigma (x_{-})$, still in negative–Cartan–positive block order. Thus $\pi \sigma ={\sigma }_{\mathfrak{h}}\pi$, where ${\sigma }_{\mathfrak{h}}$ fixes $L_0,C$, and ${\chi }_{c,h}{\sigma }_{\mathfrak{h}}={\chi }_{c,h}$. It follows that $⟨x{v}_h,y{v}_h⟩=⟨y{v}_h,x{v}_h⟩$.

For a negative generator $L_{-n}$, the first argument $L_{-n}x{v}_h$ is already in $U({\mathfrak{n}}_{-})v_h$, and the form formula gives $⟨L_{-n}x{v}_h,y{v}_h⟩={\chi }_{c,h}(\pi (\sigma (x)L_{n}y))$. To evaluate $⟨x{v}_h,L_{n}y{v}_h⟩$, put $L_{n}y$ in PBW order: terms with a positive right block vanish on $v_h$ and also have zero Harish–Chandra projection after multiplication by the left positive block $\sigma (x)$; terms with no positive right block have their Cartan factor evaluated by the same character ${\chi }_{c,h}$. Thus these two expressions agree. The projection $\pi$ is homogeneous for $\mathrm{deg}{L}_n=-n,\mathrm{deg}{L}_0=\mathrm{deg}C=0$, so step 2.1's formula pairs negative-mode monomials of grades $N,M$ to zero unless $N=M$. On an equal-grade pair, $L_0$ acts by $h+N$ in either argument and $C$ by $c$; this proves contravariance for $L_0,C$. Symmetry from step 3.1 then gives the identity for positive generators by interchanging the two arguments. The form is therefore contravariant for every generator and hence for all of $U(\mathrm{V}\mathrm{i}\mathrm{r})$.

The form pairs different $L_0$-grades to zero by the homogeneous projection argument in step 4.1. Its radical is consequently a graded submodule, since contravariance in step 4.1 moves each generator across the pairing. It is proper because the grade-zero vector has norm $1$.

This normalized contravariant form is unique. For any other such form $B^{\mathrm{\prime }}$, repeated contravariance moves a negative-mode monomial $x$ in the first argument to $\sigma (x)$ in the second: $B^{\mathrm{\prime }}(x{v}_h,y{v}_h)=B^{\mathrm{\prime }}(v_h,\sigma (x)y{v}_h)$. Put $\sigma (x)y$ in the PBW order of step 1.1. Positive right-block terms kill $v_h$; a Cartan factor acts by ${\chi }_{c,h}$; every nonempty negative block has positive grade and is orthogonal to $v_h$, since contravariance under $L_0$ gives the same grade-orthogonality calculation as in step 5.1. Thus only ${\chi }_{c,h}(\pi (\sigma (x)y))$ remains, which is the defining formula in step 2.1. So $B^{\mathrm{\prime }}$ agrees with the constructed form on a basis and hence everywhere.

Every submodule is graded: each of its vectors has finitely many homogeneous components, and polynomial interpolation in $L_0$ projects that vector onto any one of its finitely many distinct eigenvalues. A proper submodule therefore has no grade-zero vector, since the grade-zero space is $\mathbb{C}{v}_h$ and containing $v_h$ would generate all of $M(c,h)$. If $w$ lies in such a submodule and $y=L_{-{\lambda }_1}\cdots L_{-{\lambda }_k}{v}_h$ is a grade-matched PBW vector, repeated contravariance gives $⟨w,y⟩=⟨L_{{\lambda }_k}\cdots L_{{\lambda }_1}w,v_h⟩=0$: the first entry remains in the submodule and has grade zero, so it is zero. Other-grade pairings vanish by step 5.1. Thus every proper submodule lies in the radical. Since the radical itself is proper, it is the unique maximal proper submodule.